From 560cbc786fe3f0638836a4e52d8762b2ca568147 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 10 Aug 2022 22:48:43 +0800
Subject: [PATCH 01/42] 34212: clean version

---
 src/sage/rings/number_field/number_field.py | 67 +++++++++++++++++++++
 1 file changed, 67 insertions(+)

diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index 58463d570d4..3b1bb23205b 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -153,6 +153,10 @@
 from collections import Counter
 from builtins import zip
 
+from sage.categories.homset import Hom
+from sage.categories.sets_cat import Sets
+from sage.modules.free_module import VectorSpace
+from sage.modules.free_module_element import vector
 
 _NumberFields = NumberFields()
 
@@ -9297,6 +9301,69 @@ def minkowski_embedding(self, B=None, prec=None):
 
         return sage.matrix.all.matrix(d)
 
+    def logarithmic_embedding(self, prec=53):
+        """
+        Return the morphism of ``self`` under the logarithmic embedding
+        in the category Set.
+
+        The logarithmic embedding is defined as a map from the number field ``self`` to `\RR^n`.
+
+        It is defined under Definition 4.9.6 in [Cohen1993]_.
+
+        INPUT:
+
+        - ``prec`` -- desired floating point precision.
+
+        OUTPUT:
+
+        - a tuple of real numbers.
+
+        EXAMPLES::
+
+            sage: CF.<a> = CyclotomicField(97)
+            sage: f = CF.logarithmic_embedding()
+            sage: f(0)
+            (-1)
+            sage: f(7)
+            (1.94591014905531)
+
+        ::
+
+            sage: K.<a> = NumberField(x^3 + 5)
+            sage: f = K.logarithmic_embedding()
+            sage: f(0)
+            (-1)
+            sage: f(7)
+            (1.94591014905531)
+        """
+        def closure_map(x, prec=53):
+            """
+            The function closure of the logarithmic embedding.
+            """
+            K = self.base_ring()
+            K_embeddings = K.places(prec)
+            r1, r2 = K.signature()
+            r = r1 + r2 - 1
+
+            from sage.rings.all import RealField
+            Reals = RealField(prec)
+
+            if x == 0:
+                return vector([-1 for _ in range(r + 1)])
+
+            x_logs = []
+            for i in range(r1):
+                sigma = K_embeddings[i]
+                x_logs.append(Reals(abs(sigma(x))).log())
+            for i in range(r1, r + 1):
+                tau = K_embeddings[i]
+                x_logs.append(2 * Reals(abs(tau(x))).log())
+
+            return vector(x_logs)
+
+        log_map = closure_map(self(0), prec)
+        return Hom(self, VectorSpace(QQ, len(log_map)), Sets())
+
     def places(self, all_complex=False, prec=None):
         r"""
         Return the collection of all infinite places of self.

From 2ad46295eef275961547bc0e68a8cae5abb25df5 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 11 Aug 2022 14:00:41 +0800
Subject: [PATCH 02/42] 34212: fix doc

---
 src/sage/rings/number_field/number_field.py | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index 3b1bb23205b..c5a591de1ff 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -157,6 +157,7 @@
 from sage.categories.sets_cat import Sets
 from sage.modules.free_module import VectorSpace
 from sage.modules.free_module_element import vector
+from sage.rings.real_mpfr import RR
 
 _NumberFields = NumberFields()
 
@@ -9302,7 +9303,7 @@ def minkowski_embedding(self, B=None, prec=None):
         return sage.matrix.all.matrix(d)
 
     def logarithmic_embedding(self, prec=53):
-        """
+        r"""
         Return the morphism of ``self`` under the logarithmic embedding
         in the category Set.
 
@@ -9362,7 +9363,8 @@ def closure_map(x, prec=53):
             return vector(x_logs)
 
         log_map = closure_map(self(0), prec)
-        return Hom(self, VectorSpace(QQ, len(log_map)), Sets())
+        hom = Hom(self, VectorSpace(RR, len(log_map)), Sets())
+        return hom(log_map)
 
     def places(self, all_complex=False, prec=None):
         r"""

From 6bff6a6e28b8ee186fc6bc800acb692a6c7eb967 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 11 Aug 2022 16:43:53 +0800
Subject: [PATCH 03/42] 34212: fix doc and code

---
 src/sage/rings/number_field/number_field.py | 19 +++++++++----------
 1 file changed, 9 insertions(+), 10 deletions(-)

diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index c5a591de1ff..0d2f56a1dc7 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -9317,31 +9317,31 @@ def logarithmic_embedding(self, prec=53):
 
         OUTPUT:
 
-        - a tuple of real numbers.
+        - the morphism of ``self`` under the logarithmic embedding in the category Set.
 
         EXAMPLES::
 
-            sage: CF.<a> = CyclotomicField(97)
+            sage: CF.<a> = CyclotomicField(5)
             sage: f = CF.logarithmic_embedding()
             sage: f(0)
-            (-1)
+            (-1, -1)
             sage: f(7)
-            (1.94591014905531)
+            (3.89182029811063, 3.89182029811063)
 
         ::
 
             sage: K.<a> = NumberField(x^3 + 5)
             sage: f = K.logarithmic_embedding()
             sage: f(0)
-            (-1)
+            (-1, -1)
             sage: f(7)
-            (1.94591014905531)
+            (1.94591014905531, 3.89182029811063)
         """
         def closure_map(x, prec=53):
             """
             The function closure of the logarithmic embedding.
             """
-            K = self.base_ring()
+            K = self
             K_embeddings = K.places(prec)
             r1, r2 = K.signature()
             r = r1 + r2 - 1
@@ -9362,9 +9362,8 @@ def closure_map(x, prec=53):
 
             return vector(x_logs)
 
-        log_map = closure_map(self(0), prec)
-        hom = Hom(self, VectorSpace(RR, len(log_map)), Sets())
-        return hom(log_map)
+        hom = Hom(self, VectorSpace(RR, len(closure_map(self(0), prec))), Sets())
+        return hom(closure_map)
 
     def places(self, all_complex=False, prec=None):
         r"""

From 15cb0846b3348e3fc7baf3175deea6fc9e625691 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 1 Sep 2022 13:35:39 +0800
Subject: [PATCH 04/42] 34212: Correct doc and add an example

---
 src/sage/rings/number_field/number_field.py | 11 ++++++++++-
 1 file changed, 10 insertions(+), 1 deletion(-)

diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index 0d2f56a1dc7..edfa01cd232 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -9309,7 +9309,7 @@ def logarithmic_embedding(self, prec=53):
 
         The logarithmic embedding is defined as a map from the number field ``self`` to `\RR^n`.
 
-        It is defined under Definition 4.9.6 in [Cohen1993]_.
+        It is defined under Definition 4.9.6 in [Coh1993]_.
 
         INPUT:
 
@@ -9336,6 +9336,15 @@ def logarithmic_embedding(self, prec=53):
             (-1, -1)
             sage: f(7)
             (1.94591014905531, 3.89182029811063)
+
+        ::
+
+            sage: K.<a> = NumberField(x^4 - 8*x^2 + 3)
+            sage: f = logarithmic_embedding(K)
+            sage: f(0)
+            (-1, -1, -1, -1)
+            sage: f(7)
+            (3.89182029811063, 3.89182029811063, 3.89182029811063, 3.89182029811063)
         """
         def closure_map(x, prec=53):
             """

From 89d2c29e89e8cacd90d85bab0b7cbe9556e8614d Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 1 Sep 2022 13:42:27 +0800
Subject: [PATCH 05/42] 34212: log embedding for relative number field

---
 .../rings/number_field/number_field_rel.py    | 69 +++++++++++++++++++
 1 file changed, 69 insertions(+)

diff --git a/src/sage/rings/number_field/number_field_rel.py b/src/sage/rings/number_field/number_field_rel.py
index d33980c4b10..499a89161a6 100644
--- a/src/sage/rings/number_field/number_field_rel.py
+++ b/src/sage/rings/number_field/number_field_rel.py
@@ -2062,6 +2062,75 @@ def automorphisms(self):
                                         check=False, universe=self.Hom(self))
         return self.__automorphisms
 
+    def logarithmic_embedding(self, prec=53):
+        r"""
+        Return the morphism of ``self`` under the logarithmic embedding
+        in the category Set.
+
+        The logarithmic embedding is defined as a map from the relative number field ``self`` to `\RR^n`.
+
+        It is defined under Definition 4.9.6 in [Coh1993]_.
+
+        INPUT:
+
+        - ``prec`` -- desired floating point precision.
+
+        OUTPUT:
+
+        - the morphism of ``self`` under the logarithmic embedding in the category Set.
+
+        EXAMPLES::
+
+            sage: K.<k> = CyclotomicField(3)
+            sage: R.<x> = K[]
+            sage: L.<l> = K.extension(x^5 + 5)
+            sage: f = logarithmic_embedding(L)
+            sage: f(0)
+            (-1, -1, -1, -1, -1)
+            sage: f(5)
+            (3.21887582486820, 3.21887582486820, 3.21887582486820,
+            3.21887582486820, 3.21887582486820)
+
+        ::
+
+            sage: K.<i> = NumberField(x^2 + 1)
+            sage: t = K['t'].gen()
+            sage: L.<a> = K.extension(t^4 - i)
+            sage: f = logarithmic_embedding(L)
+            sage: f(0)
+            (-1, -1, -1, -1, -1, -1, -1, -1)
+            sage: f(3)
+            (2.19722457733622, 2.19722457733622, 2.19722457733622, 2.19722457733622,
+            2.19722457733622, 2.19722457733622, 2.19722457733622, 2.19722457733622)
+        """
+        def closure_map(x, prec=53):
+            """
+            The function closure of the logarithmic embedding.
+            """
+            K = self
+            K_embeddings = K.places(prec)
+            r1, r2 = K.signature()
+            r = r1 + r2 - 1
+
+            from sage.rings.all import RealField
+            Reals = RealField(prec)
+
+            if x == 0:
+                return vector([-1 for _ in range(r + 1)])
+
+            x_logs = []
+            for i in range(r1):
+                sigma = K_embeddings[i]
+                x_logs.append(Reals(abs(sigma(x))).log())
+            for i in range(r1, r + 1):
+                tau = K_embeddings[i]
+                x_logs.append(2 * Reals(abs(tau(x))).log())
+
+            return vector(x_logs)
+
+        hom = Hom(self, VectorSpace(RR, len(closure_map(self(0), prec))), Sets())
+        return hom(closure_map)
+
     def places(self, all_complex=False, prec=None):
         """
         Return the collection of all infinite places of self.

From 2fd483097aaa8d2a849bcb6785fc7f967ef73909 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 1 Sep 2022 14:51:33 +0800
Subject: [PATCH 06/42] 34212: correct examples

---
 src/sage/rings/number_field/number_field.py     |  6 +++---
 src/sage/rings/number_field/number_field_rel.py | 12 ++++++++----
 2 files changed, 11 insertions(+), 7 deletions(-)

diff --git a/src/sage/rings/number_field/number_field.py b/src/sage/rings/number_field/number_field.py
index edfa01cd232..118386623c3 100644
--- a/src/sage/rings/number_field/number_field.py
+++ b/src/sage/rings/number_field/number_field.py
@@ -9339,12 +9339,12 @@ def logarithmic_embedding(self, prec=53):
 
         ::
 
-            sage: K.<a> = NumberField(x^4 - 8*x^2 + 3)
-            sage: f = logarithmic_embedding(K)
+            sage: F.<a> = NumberField(x^4 - 8*x^2 + 3)
+            sage: f = F.logarithmic_embedding()
             sage: f(0)
             (-1, -1, -1, -1)
             sage: f(7)
-            (3.89182029811063, 3.89182029811063, 3.89182029811063, 3.89182029811063)
+            (1.94591014905531, 1.94591014905531, 1.94591014905531, 1.94591014905531)
         """
         def closure_map(x, prec=53):
             """
diff --git a/src/sage/rings/number_field/number_field_rel.py b/src/sage/rings/number_field/number_field_rel.py
index 499a89161a6..9e80ef1e3c1 100644
--- a/src/sage/rings/number_field/number_field_rel.py
+++ b/src/sage/rings/number_field/number_field_rel.py
@@ -99,8 +99,12 @@
 from sage.rings.number_field.morphism import RelativeNumberFieldHomomorphism_from_abs
 from sage.libs.pari.all import pari_gen
 
-from sage.rings.rational_field import QQ
-from sage.rings.integer_ring import ZZ
+from sage.categories.homset import Hom
+from sage.categories.sets_cat import Sets
+from sage.modules.free_module import VectorSpace
+from sage.modules.free_module_element import vector
+
+from sage.rings.all import RR, QQ, ZZ
 
 
 def is_RelativeNumberField(x):
@@ -2084,7 +2088,7 @@ def logarithmic_embedding(self, prec=53):
             sage: K.<k> = CyclotomicField(3)
             sage: R.<x> = K[]
             sage: L.<l> = K.extension(x^5 + 5)
-            sage: f = logarithmic_embedding(L)
+            sage: f = L.logarithmic_embedding()
             sage: f(0)
             (-1, -1, -1, -1, -1)
             sage: f(5)
@@ -2096,7 +2100,7 @@ def logarithmic_embedding(self, prec=53):
             sage: K.<i> = NumberField(x^2 + 1)
             sage: t = K['t'].gen()
             sage: L.<a> = K.extension(t^4 - i)
-            sage: f = logarithmic_embedding(L)
+            sage: f = L.logarithmic_embedding()
             sage: f(0)
             (-1, -1, -1, -1, -1, -1, -1, -1)
             sage: f(3)

From 7a12ff3327a209bc6065fb0d8ea9b939f1d91c5f Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 13:28:09 +0800
Subject: [PATCH 07/42] points

---
 src/doc/en/reference/references/index.rst     |   4 +
 .../schemes/projective/proj_bdd_height.py     | 189 ++++++++++++++++++
 2 files changed, 193 insertions(+)
 create mode 100644 src/sage/schemes/projective/proj_bdd_height.py

diff --git a/src/doc/en/reference/references/index.rst b/src/doc/en/reference/references/index.rst
index 1dbd6f036e7..010654f683a 100644
--- a/src/doc/en/reference/references/index.rst
+++ b/src/doc/en/reference/references/index.rst
@@ -3668,6 +3668,10 @@ REFERENCES:
             and its use for certified homotopy continuation of systems of plane
             algebraic curves, :arxiv:`1505.03432`
 
+.. [Krumm2016] Daid Krumm, *Computing Points of Bounded Height in Projective Space over a Number Field*,
+               MATHEMATICS OF COMPUTATION, Volume 85, Number 297, January 2016, Pages 423–447.
+               http://dx.doi.org/10.1090/mcom/2984
+
 .. [KR2001] \J. Kahane and A. Ryba. *The hexad game*, Electronic
             Journal of Combinatorics, **8**
             (2001). http://www.combinatorics.org/Volume_8/Abstracts/v8i2r11.html
diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
new file mode 100644
index 00000000000..8da3d90297f
--- /dev/null
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -0,0 +1,189 @@
+r"""
+Points of bounded height in projective spaces
+
+This module defines sunctions to compute points of bounded height of a given
+number field with height less than a specified bound in projective spaces.
+
+Sage functions to list all elements of a given number field with height less
+than a specified bound.
+
+AUTHORS:
+
+- Jing Guo (2022): initial version based on David Krumm's code
+
+REFERENCES:
+
+- [Krumm2016]
+
+"""
+
+import itertools
+
+def QQ_points_of_bounded_height(dim, bound):
+    r"""
+    ...
+    """
+    if bound < 1:
+        return
+    PN = ProjectiveSpace(QQ, dim)
+    unit_tuples = list(itertools.product([-1, 1], repeat=dim))
+    increasing_tuples = itertools.combinations_with_replacement(range(bound + 1), dim + 1)
+    for t in increasing_tuples:
+        if gcd(t) == 1:
+            for p in itertools.permutations(t):
+                for u in unit_tuples:
+                    new_point = [a*b for a, b in zip(u, p)] + [p[dim]]
+                    yield PN(new_point)
+
+def points_of_bounded_height(K, dim, bound, prec=53):
+    r"""
+    ...
+    """
+    if bound < 1:
+        return
+
+    r1, r2 = K.signature()
+    r = r1 + r2 - 1
+    K_degree = K.degree() 
+    K_embeddings = K.places(prec=prec)
+    roots_of_unity = K.roots_of_unity()
+    unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
+
+    PN = ProjectiveSpace(K, dim)
+
+    Reals = RealField(prec)
+    logB = Reals(bound).log()
+
+    class_group_ideals = [c.ideal() for c in K.class_group()]
+    class_number = len(class_group_ideals)
+    class_group_ideal_norms = [a.norm() for a in class_group_ideals]
+    norm_bound = bound*max(class_group_ideal_norms)
+    fundamental_units = UnitGroup(K).fundamental_units()
+    fund_unit_logs = map(log_map, fundamental_units) # TODO log map
+
+    test_matrix = column_matrix(fund_unit_logs)
+    try:
+        test_matrix.change_ring(QQ)
+    except ValueError:
+        raise ValueError('prec too low.')
+
+    cut_fund_unit_logs = column_matrix(fund_unit_logs).delete_rows([r])
+    lll_fund_units = []
+    for c in pari(cut_fund_unit_logs).qflll().python().columns():
+        new_unit = 1
+        for i in range(r):
+            new_unit *= fundamental_units[i]**c[i]
+        lll_fund_units.append(new_unit)
+    fundamental_units = lll_fund_units
+    fund_unit_logs = map(log_map, fundamental_units)
+
+    possible_norm_set = set([])
+    for i in range(class_number):
+        for k in range(1, bound + 1):
+            possible_norm_set.add(k*class_group_ideal_norms[i])
+    
+    principal_ideal_gens = dict()
+    negative_norm_units = K.elements_of_norm(-1)
+    if len(negative_norm_units) == 0:
+        for m in possible_norm_set:
+            principal_ideal_gens[m] = K.elements_of_norm(m) + K.elements_of_norm(-m)
+    else:
+        for m in possible_norm_set:
+            principal_ideal_gens[m] = K.elements_of_norm(m)
+
+    pr_ideal_gen_logs = dict()
+    for key in principal_ideal_gens:
+        for y in principal_ideal_gens[key]:
+            pr_ideal_gen_logs[y] = log_map(y)
+    
+    fund_parallelotope_vertices = []
+    for coefficient_tuple in itertools.product([-1/2, 1/2], repeat=r):
+        vertex = sum([coefficient_tuple[i]*fund_unit_logs[i] for i in range(r)])
+        fund_parallelotope_vertices.append(vertex)
+      
+    D_numbers = []
+    for v in range(r + 1):
+        D_numbers.append(max([vertex[v] for vertex in fund_parallelotope_vertices]))
+    
+    A_numbers = []
+    for v in range(r + 1):
+        A_numbers.append(min([pr_ideal_gen_logs[y][v] for y in pr_ideal_gen_logs]))
+    
+    aux_constant = (1/K_degree)*Reals(norm_bound).log()
+    
+    L_numbers = []
+    for v in range(r1):
+        L_numbers.append(aux_constant + D_numbers[v] - A_numbers[v])
+    for v in range(r1, r + 1):
+        L_numbers.append(2*aux_constant + D_numbers[v] - A_numbers[v])
+    L_numbers = vector(L_numbers).change_ring(QQ)
+
+    T = column_matrix(fund_unit_logs).delete_rows([r]).change_ring(QQ)
+
+    M = ((-1)*matrix.identity(r)).insert_row(r, [1 for i in range(r)])
+    M = M.transpose().insert_row(0, [0 for i in range(r + 1)]).transpose()
+    M = M.change_ring(QQ)
+    M.set_column(0, L_numbers)
+    vertices = map(vector, Polyhedron(ieqs=list(M)).vertices())
+
+    T_it = T.inverse().transpose()
+    unit_polytope = Polyhedron([v*T_it for v in vertices])
+
+    coordinate_space = dict()
+    coordinate_space[0] = [[K(0), log_map(0)]]
+    int_points = unit_polytope.integral_points()
+
+    units_with_logs = dict()
+    for n in int_points:
+        new_unit = 1
+        for j in range(r):
+            new_unit *= fundamental_units[j]**n[j]
+        new_unit_log = sum([n[j]*fund_unit_logs[j] for j in range(r)])
+        units_with_logs[n] = [new_unit, new_unit_log]
+
+    for norm in principal_ideal_gens:
+        coordinate_list = []
+        for y in principal_ideal_gens[norm]:
+            for n in int_points:
+                unit, unit_log = units_with_logs[n]
+                y_log = pr_ideal_gen_logs[y]
+                g_log = unit_log + y_log
+                bool1 = all(g_log[i] <= aux_constant + D_numbers[i] for i in range(r1))
+                bool2 = all(g_log[j] <= 2*aux_constant + D_numbers[j] for j in range(r1, r + 1))
+                if bool1 and bool2:
+                    g = unit*y
+                    coordinate_list.append([g, g_log])
+        if len(coordinate_list) > 0:
+            coordinate_space[norm] = coordinate_list
+
+    for m in range(class_number):
+        a = class_group_ideals[m]
+        a_norm = class_group_ideal_norms[m]
+        log_a_norm = Reals(a_norm).log()
+        a_const = (logB + log_a_norm)/K_degree
+        a_coordinates = []
+
+        for k in range(bound + 1):
+            norm = k*a_norm
+            if coordinate_space.has_key(norm):
+                for pair in coordinate_space[norm]:
+                    g, g_log = pair
+                    if g in a:
+                        bool1 = all(g_log[i] <= a_const + D_numbers[i] for i in range(r1))
+                        bool2 = all(g_log[j] <= 2*a_const + D_numbers[j] for j in range(r1, r + 1))
+                        if bool1 and bool2:
+                            a_coordinates.append(pair)    
+
+        t = len(a_coordinates) - 1
+        points_in_class_a = set([])
+        increasing_tuples = itertools.combinations_with_replacement(range(t + 1), dim + 1)
+        log_arch_height_bound = logB + log_a_norm
+        for index_tuple in increasing_tuples:
+            point_coordinates = [a_coordinates[i][0] for i in index_tuple]
+            point_coordinate_logs = [a_coordinates[i][1] for i in index_tuple]
+            log_arch_height = sum([max([x[i] for x in point_coordinate_logs]) for i in range(r + 1)])
+            if log_arch_height <= log_arch_height_bound and a == K.ideal(point_coordinates):
+                for p in itertools.permutations(point_coordinates):
+                    for u in unit_tuples:
+                        new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
+                        yield PN(new_point)

From 2ed790e656bfe439dd7cdfe49c4c5ad45488b5f5 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 15:50:20 +0800
Subject: [PATCH 08/42] clean

---
 .../schemes/projective/projective_space.py    | 74 ++++++-------------
 1 file changed, 24 insertions(+), 50 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index ed44ebcd6aa..55a2f7e6a5b 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1852,18 +1852,12 @@ def _morphism(self, *args, **kwds):
 
     def points_of_bounded_height(self, **kwds):
         r"""
-        Returns an iterator of the points in self of absolute height of at most the given bound.
+        Return an iterator of the points in ``self`` of absolute height of
+        at most the given bound.
 
-        Bound check is strict for the rational field. Requires self to be projective space
-        over a number field. Uses the
-        Doyle-Krumm algorithm 4 (algorithm 5 for imaginary quadratic) for
-        computing algebraic numbers up to a given height [DK2013]_.
+        ALGORITHM:
 
-        The algorithm requires floating point arithmetic, so the user is
-        allowed to specify the precision for such calculations.
-        Additionally, due to floating point issues, points
-        slightly larger than the bound may be returned. This can be controlled
-        by lowering the tolerance.
+        This is an implementation of Algorithm 6 in [Krumm2016]_.
 
         INPUT:
 
@@ -1871,13 +1865,11 @@ def points_of_bounded_height(self, **kwds):
 
         - ``bound`` - a real number
 
-        - ``tolerance`` - a rational number in (0,1] used in doyle-krumm algorithm-4
-
-        - ``precision`` - the precision to use for computing the elements of bounded height of number fields.
+        - ``precision`` - (default: 53) a positive integer
 
         OUTPUT:
 
-        - an iterator of points in this space
+        - an iterator of points of bounded height
 
         EXAMPLES::
 
@@ -1897,48 +1889,30 @@ def points_of_bounded_height(self, **kwds):
             sage: len(list(P.points_of_bounded_height(bound=1.5, tolerance=0.1)))
             57
         """
-        if is_RationalField(self.base_ring()):
-            ftype = False  # stores whether the field is a number field or the rational field
-        elif self.base_ring() in NumberFields():  # true for rational field as well, so check is_RationalField first
-            ftype = True
+        from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, points_of_bounded_height
+
+        R = self.base_ring()
+
+        # whether the field is a number field or the rational field
+        if is_RationalField(R):
+            field_type = False
+        elif R in NumberFields():
+            # true for rational field as well, so check is_RationalField first
+            field_type = True
         else:
             raise NotImplementedError("self must be projective space over a number field")
 
         bound = kwds.pop('bound')
         B = bound**(self.base_ring().absolute_degree())  # convert to relative height
 
-        n = self.dimension_relative()
-        R = self.base_ring()
-        if ftype:
-            zero = R.zero()
-            i = n
-            while not i < 0:
-                P = [zero for _ in range(i)] + [R.one()]
-                P += [zero for _ in range(n - i)]
-                yield self(P)
-                tol = kwds.pop('tolerance', 1e-2)
-                prec = kwds.pop('precision', 53)
-                iters = [R.elements_of_bounded_height(bound=B, tolerance=tol, precision=prec) for _ in range(i)]
-                for x in iters:
-                    next(x)  # put at zero
-                j = 0
-                while j < i:
-                    try:
-                        P[j] = next(iters[j])
-                        yield self(P)
-                        j = 0
-                    except StopIteration:
-                        iters[j] = R.elements_of_bounded_height(bound=B, tolerance=tol, precision=prec)  # reset
-                        next(iters[j])  # put at zero
-                        P[j] = zero
-                        j += 1
-                i -= 1
-        else:  # base ring QQ
-            zero = (0,) * (n + 1)
-            for c in cartesian_product_iterator([srange(-B, B + 1)
-                                                 for _ in range(n + 1)]):
-                if gcd(c) == 1 and c > zero:
-                    yield self.point(c, check=False)
+        prec = kwds.pop('precision', 53)
+
+        dim = self.dimension_relative()
+
+        if field_type:
+            return points_of_bounded_height(dim, B)
+        else:
+            return QQ_points_of_bounded_height(R, dim, B, prec)
 
     def subscheme_from_Chow_form(self, Ch, dim):
         r"""

From b1b8a28be151beff6455102b2fb0e0f4b564bde8 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 16:10:09 +0800
Subject: [PATCH 09/42] clean

---
 .../schemes/projective/proj_bdd_height.py     | 36 +++++++++++++++++--
 1 file changed, 34 insertions(+), 2 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 8da3d90297f..866453ba2a7 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -18,10 +18,23 @@
 """
 
 import itertools
+from sage.schemes.projective.projective_space import ProjectiveSpace
+
 
 def QQ_points_of_bounded_height(dim, bound):
     r"""
-    ...
+    Return an iterator of the points in ``self`` of absolute height of
+    at most the given bound in the rational field.
+
+    INPUT:
+
+    - ``dim`` -- a positive interger
+
+    - ``bound`` -- a real number
+
+    OUTPUT:
+
+    - an iterator of points of bounded height
     """
     if bound < 1:
         return
@@ -37,7 +50,26 @@ def QQ_points_of_bounded_height(dim, bound):
 
 def points_of_bounded_height(K, dim, bound, prec=53):
     r"""
-    ...
+    Return an iterator of the points in ``self`` of absolute height of
+    at most the given bound.
+
+    ALGORITHM:
+
+    This is an implementation of Algorithm 6 in [Krumm2016]_.
+
+    INPUT:
+
+    - ``K`` -- a number field
+
+    - ``dim`` -- a positive interger
+
+    - ``bound`` -- a real number
+
+    - ``prec`` -- (default: 53) a positive integer
+
+    OUTPUT:
+
+    - an iterator of points of bounded height
     """
     if bound < 1:
         return

From 76dab1c5ae1a007e5fd4e02301269ea6c46d237a Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 17:25:07 +0800
Subject: [PATCH 10/42] rational

---
 .../schemes/projective/proj_bdd_height.py     | 27 +++++++++++++++----
 .../schemes/projective/projective_space.py    | 23 +++++-----------
 2 files changed, 29 insertions(+), 21 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 866453ba2a7..1b37bb9724b 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -19,34 +19,51 @@
 
 import itertools
 from sage.schemes.projective.projective_space import ProjectiveSpace
-
+from sage.rings.rational_field import QQ
+from sage.arith.all import gcd
 
 def QQ_points_of_bounded_height(dim, bound):
     r"""
     Return an iterator of the points in ``self`` of absolute height of
-    at most the given bound in the rational field.
+    at most ``bound`` in the rational field.
 
     INPUT:
 
-    - ``dim`` -- a positive interger
+    - ``dim`` -- a positive integer
 
     - ``bound`` -- a real number
 
     OUTPUT:
 
     - an iterator of points of bounded height
+
+    EXAMPLES:
+
+        sage: from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height
+        sage: list(QQ_points_of_bounded_height(1, 1))
+        [(0 : 1), (1 : 0), (1 : 1), (-1 : 1)]
+        sage: len(list(QQ_points_of_bounded_height(1, 5)))
+        40
+
+    There are no points of negative height::
+
+        sage: from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height
+        sage: list(QQ_points_of_bounded_height(1, -3))
+        []
     """
     if bound < 1:
-        return
+        return iter(set([]))
     PN = ProjectiveSpace(QQ, dim)
     unit_tuples = list(itertools.product([-1, 1], repeat=dim))
+    points_of_bounded_height = set([])
     increasing_tuples = itertools.combinations_with_replacement(range(bound + 1), dim + 1)
     for t in increasing_tuples:
         if gcd(t) == 1:
             for p in itertools.permutations(t):
                 for u in unit_tuples:
                     new_point = [a*b for a, b in zip(u, p)] + [p[dim]]
-                    yield PN(new_point)
+                    points_of_bounded_height.add(PN(new_point))
+    return iter(points_of_bounded_height)
 
 def points_of_bounded_height(K, dim, bound, prec=53):
     r"""
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 55a2f7e6a5b..93b8d8c7b8a 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1874,20 +1874,11 @@ def points_of_bounded_height(self, **kwds):
         EXAMPLES::
 
             sage: P.<x,y> = ProjectiveSpace(QQ, 1)
-            sage: sorted(list(P.points_of_bounded_height(bound=5)))
-            [(0 : 1), (1 : -5), (1 : -4), (1 : -3), (1 : -2), (1 : -1), (1 : 0),
-             (1 : 1), (1 : 2), (1 : 3), (1 : 4), (1 : 5), (2 : -5), (2 : -3),
-             (2 : -1), (2 : 1), (2 : 3), (2 : 5), (3 : -5), (3 : -4), (3 : -2),
-             (3 : -1), (3 : 1), (3 : 2), (3 : 4), (3 : 5), (4 : -5), (4 : -3),
-             (4 : -1), (4 : 1), (4 : 3), (4 : 5), (5 : -4), (5 : -3), (5 : -2),
-             (5 : -1), (5 : 1), (5 : 2), (5 : 3), (5 : 4)]
-
-        ::
-
-            sage: u = QQ['u'].0
-            sage: P.<x,y,z> = ProjectiveSpace(NumberField(u^2 - 2, 'v'), 2)
-            sage: len(list(P.points_of_bounded_height(bound=1.5, tolerance=0.1)))
-            57
+            sage: sorted(list(P.points_of_bounded_height(bound=3)))
+            [(-3 : 1), (-2 : 1), (-3/2 : 1), (-1 : 1),
+             (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (0 : 1),
+             (1/3 : 1), (1/2 : 1), (2/3 : 1), (1 : 0),
+             (1 : 1), (3/2 : 1), (2 : 1), (3 : 1)]
         """
         from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, points_of_bounded_height
 
@@ -1910,9 +1901,9 @@ def points_of_bounded_height(self, **kwds):
         dim = self.dimension_relative()
 
         if field_type:
-            return points_of_bounded_height(dim, B)
+            return points_of_bounded_height(R, dim, B, prec)
         else:
-            return QQ_points_of_bounded_height(R, dim, B, prec)
+            return QQ_points_of_bounded_height(dim, B)
 
     def subscheme_from_Chow_form(self, Ch, dim):
         r"""

From 706e514ae198f296bf29808785cbc2add3b89c5b Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 17:43:39 +0800
Subject: [PATCH 11/42] import

---
 src/sage/schemes/projective/proj_bdd_height.py | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 1b37bb9724b..5f637d98f9b 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -20,6 +20,8 @@
 import itertools
 from sage.schemes.projective.projective_space import ProjectiveSpace
 from sage.rings.rational_field import QQ
+from sage.rings.all import RealField
+from sage.rings.number_field.unit_group import UnitGroup
 from sage.arith.all import gcd
 
 def QQ_points_of_bounded_height(dim, bound):

From 3e5f25f3015a452b6da3669753b5336457e4b8b7 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 18:05:42 +0800
Subject: [PATCH 12/42] number fields

---
 src/sage/schemes/projective/proj_bdd_height.py | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 5f637d98f9b..07ce8d5f959 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -23,6 +23,7 @@
 from sage.rings.all import RealField
 from sage.rings.number_field.unit_group import UnitGroup
 from sage.arith.all import gcd
+from sage.rings.number_field.number_field import logarithmic_embedding
 
 def QQ_points_of_bounded_height(dim, bound):
     r"""
@@ -110,7 +111,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     class_group_ideal_norms = [a.norm() for a in class_group_ideals]
     norm_bound = bound*max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
-    fund_unit_logs = map(log_map, fundamental_units) # TODO log map
+    fund_unit_logs = map(logarithmic_embedding, fundamental_units)
 
     test_matrix = column_matrix(fund_unit_logs)
     try:
@@ -126,7 +127,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
             new_unit *= fundamental_units[i]**c[i]
         lll_fund_units.append(new_unit)
     fundamental_units = lll_fund_units
-    fund_unit_logs = map(log_map, fundamental_units)
+    fund_unit_logs = map(logarithmic_embedding, fundamental_units)
 
     possible_norm_set = set([])
     for i in range(class_number):
@@ -145,7 +146,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     pr_ideal_gen_logs = dict()
     for key in principal_ideal_gens:
         for y in principal_ideal_gens[key]:
-            pr_ideal_gen_logs[y] = log_map(y)
+            pr_ideal_gen_logs[y] = logarithmic_embedding(y)
     
     fund_parallelotope_vertices = []
     for coefficient_tuple in itertools.product([-1/2, 1/2], repeat=r):
@@ -181,7 +182,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     unit_polytope = Polyhedron([v*T_it for v in vertices])
 
     coordinate_space = dict()
-    coordinate_space[0] = [[K(0), log_map(0)]]
+    coordinate_space[0] = [[K(0), logarithmic_embedding(0)]]
     int_points = unit_polytope.integral_points()
 
     units_with_logs = dict()

From 4d10b14e3ddbdaef3d1e2306b4e34a4255242f81 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 28 Aug 2022 19:28:41 +0800
Subject: [PATCH 13/42] log embed

---
 src/sage/schemes/projective/proj_bdd_height.py | 12 +++++++-----
 1 file changed, 7 insertions(+), 5 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 07ce8d5f959..8b2713e19da 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -23,7 +23,7 @@
 from sage.rings.all import RealField
 from sage.rings.number_field.unit_group import UnitGroup
 from sage.arith.all import gcd
-from sage.rings.number_field.number_field import logarithmic_embedding
+from sage.matrix.constructor import column_matrix
 
 def QQ_points_of_bounded_height(dim, bound):
     r"""
@@ -102,6 +102,8 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
 
     PN = ProjectiveSpace(K, dim)
+    #R = K.base_ring()
+    log_embed = K.logarithmic_embedding()
 
     Reals = RealField(prec)
     logB = Reals(bound).log()
@@ -111,7 +113,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     class_group_ideal_norms = [a.norm() for a in class_group_ideals]
     norm_bound = bound*max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
-    fund_unit_logs = map(logarithmic_embedding, fundamental_units)
+    fund_unit_logs = map(log_embed, fundamental_units)
 
     test_matrix = column_matrix(fund_unit_logs)
     try:
@@ -127,7 +129,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
             new_unit *= fundamental_units[i]**c[i]
         lll_fund_units.append(new_unit)
     fundamental_units = lll_fund_units
-    fund_unit_logs = map(logarithmic_embedding, fundamental_units)
+    fund_unit_logs = map(log_embed, fundamental_units)
 
     possible_norm_set = set([])
     for i in range(class_number):
@@ -146,7 +148,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     pr_ideal_gen_logs = dict()
     for key in principal_ideal_gens:
         for y in principal_ideal_gens[key]:
-            pr_ideal_gen_logs[y] = logarithmic_embedding(y)
+            pr_ideal_gen_logs[y] = log_embed(y)
     
     fund_parallelotope_vertices = []
     for coefficient_tuple in itertools.product([-1/2, 1/2], repeat=r):
@@ -182,7 +184,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     unit_polytope = Polyhedron([v*T_it for v in vertices])
 
     coordinate_space = dict()
-    coordinate_space[0] = [[K(0), logarithmic_embedding(0)]]
+    coordinate_space[0] = [[K(0), log_embed(0)]]
     int_points = unit_polytope.integral_points()
 
     units_with_logs = dict()

From 21c8ec68f09eb6e375c1a16a6bdbcc0507dd6fda Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Mon, 29 Aug 2022 11:25:55 +0800
Subject: [PATCH 14/42] doc

---
 src/sage/schemes/projective/proj_bdd_height.py | 5 ++---
 1 file changed, 2 insertions(+), 3 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 8b2713e19da..602dd056bf1 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -70,8 +70,8 @@ def QQ_points_of_bounded_height(dim, bound):
 
 def points_of_bounded_height(K, dim, bound, prec=53):
     r"""
-    Return an iterator of the points in ``self`` of absolute height of
-    at most the given bound.
+    Return an iterator of the points in ``K`` with dimension ``dim`` of
+    absolute height of at most ``bound``.
 
     ALGORITHM:
 
@@ -102,7 +102,6 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
 
     PN = ProjectiveSpace(K, dim)
-    #R = K.base_ring()
     log_embed = K.logarithmic_embedding()
 
     Reals = RealField(prec)

From 2711a1d38a637cf42e1cf0a12f09a2f4f42f4a9e Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Mon, 29 Aug 2022 12:10:32 +0800
Subject: [PATCH 15/42] indent

---
 .../schemes/projective/proj_bdd_height.py     | 22 ++++++++++---------
 1 file changed, 12 insertions(+), 10 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 602dd056bf1..8ad4fb6f101 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -92,11 +92,11 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     - an iterator of points of bounded height
     """
     if bound < 1:
-        return
+        return []
 
     r1, r2 = K.signature()
     r = r1 + r2 - 1
-    K_degree = K.degree() 
+    K_degree = K.degree()
     K_embeddings = K.places(prec=prec)
     roots_of_unity = K.roots_of_unity()
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
@@ -134,7 +134,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     for i in range(class_number):
         for k in range(1, bound + 1):
             possible_norm_set.add(k*class_group_ideal_norms[i])
-    
+
     principal_ideal_gens = dict()
     negative_norm_units = K.elements_of_norm(-1)
     if len(negative_norm_units) == 0:
@@ -148,22 +148,22 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     for key in principal_ideal_gens:
         for y in principal_ideal_gens[key]:
             pr_ideal_gen_logs[y] = log_embed(y)
-    
+
     fund_parallelotope_vertices = []
     for coefficient_tuple in itertools.product([-1/2, 1/2], repeat=r):
         vertex = sum([coefficient_tuple[i]*fund_unit_logs[i] for i in range(r)])
         fund_parallelotope_vertices.append(vertex)
-      
+
     D_numbers = []
     for v in range(r + 1):
         D_numbers.append(max([vertex[v] for vertex in fund_parallelotope_vertices]))
-    
+
     A_numbers = []
     for v in range(r + 1):
         A_numbers.append(min([pr_ideal_gen_logs[y][v] for y in pr_ideal_gen_logs]))
-    
+
     aux_constant = (1/K_degree)*Reals(norm_bound).log()
-    
+
     L_numbers = []
     for v in range(r1):
         L_numbers.append(aux_constant + D_numbers[v] - A_numbers[v])
@@ -225,7 +225,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
                         bool1 = all(g_log[i] <= a_const + D_numbers[i] for i in range(r1))
                         bool2 = all(g_log[j] <= 2*a_const + D_numbers[j] for j in range(r1, r + 1))
                         if bool1 and bool2:
-                            a_coordinates.append(pair)    
+                            a_coordinates.append(pair)
 
         t = len(a_coordinates) - 1
         points_in_class_a = set([])
@@ -239,4 +239,6 @@ def points_of_bounded_height(K, dim, bound, prec=53):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
                         new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
-                        yield PN(new_point)
+                        points_in_class_a.add(PN(new_point))
+        points_of_bdd_height += list(points_in_class_a)
+    return points_of_bdd_height

From 394bf27029bee8d0326999795c8cc06dc860e5bd Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Mon, 29 Aug 2022 13:33:38 +0800
Subject: [PATCH 16/42] python 2 to python 3

---
 .../schemes/projective/proj_bdd_height.py     | 34 ++++++++++++-------
 1 file changed, 22 insertions(+), 12 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 8ad4fb6f101..d814371de08 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -23,7 +23,12 @@
 from sage.rings.all import RealField
 from sage.rings.number_field.unit_group import UnitGroup
 from sage.arith.all import gcd
-from sage.matrix.constructor import column_matrix
+from sage.matrix.constructor import matrix, column_matrix
+from sage.libs.pari.all import pari
+from sage.modules.free_module_element import vector
+from sage.rings.integer import Integer
+from sage.geometry.polyhedron.constructor import Polyhedron
+
 
 def QQ_points_of_bounded_height(dim, bound):
     r"""
@@ -107,28 +112,33 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     Reals = RealField(prec)
     logB = Reals(bound).log()
 
+    points_of_bdd_height = []
+
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_number = len(class_group_ideals)
     class_group_ideal_norms = [a.norm() for a in class_group_ideals]
     norm_bound = bound*max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
-    fund_unit_logs = map(log_embed, fundamental_units)
+    fund_unit_logs = list(map(log_embed, fundamental_units))
 
     test_matrix = column_matrix(fund_unit_logs)
-    try:
-        test_matrix.change_ring(QQ)
-    except ValueError:
-        raise ValueError('prec too low.')
+    #try:
+    #    test_matrix.change_ring(QQ)
+    #except ValueError:
+    #    raise ValueError('prec too low.')
 
-    cut_fund_unit_logs = column_matrix(fund_unit_logs).delete_rows([r])
+    cut_fund_unit_logs = test_matrix.delete_rows([r])
     lll_fund_units = []
-    for c in pari(cut_fund_unit_logs).qflll().python().columns():
+    for c in cut_fund_unit_logs.columns():
+    #for c in pari(cut_fund_unit_logs).qflll().python().columns():
         new_unit = 1
         for i in range(r):
+            print(fundamental_units[i])
+            print(c[i])
             new_unit *= fundamental_units[i]**c[i]
         lll_fund_units.append(new_unit)
     fundamental_units = lll_fund_units
-    fund_unit_logs = map(log_embed, fundamental_units)
+    fund_unit_logs = list(map(log_embed, fundamental_units))
 
     possible_norm_set = set([])
     for i in range(class_number):
@@ -173,8 +183,8 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     T = column_matrix(fund_unit_logs).delete_rows([r]).change_ring(QQ)
 
-    M = ((-1)*matrix.identity(r)).insert_row(r, [1 for i in range(r)])
-    M = M.transpose().insert_row(0, [0 for i in range(r + 1)]).transpose()
+    M = ((-1)*matrix.identity(r)).insert_row(r, [Integer(1) for i in range(r)])
+    M = M.transpose().insert_row(0, [Integer(0) for i in range(r + 1)]).transpose()
     M = M.change_ring(QQ)
     M.set_column(0, L_numbers)
     vertices = map(vector, Polyhedron(ieqs=list(M)).vertices())
@@ -218,7 +228,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
         for k in range(bound + 1):
             norm = k*a_norm
-            if coordinate_space.has_key(norm):
+            if norm in coordinate_space:
                 for pair in coordinate_space[norm]:
                     g, g_log = pair
                     if g in a:

From 2e9df04ae57717bc61f2aff97f1d83490833886f Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Tue, 30 Aug 2022 20:30:33 +0800
Subject: [PATCH 17/42] some small changes

---
 .../schemes/projective/proj_bdd_height.py     | 29 ++++++++++---------
 1 file changed, 15 insertions(+), 14 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index d814371de08..e20bf297962 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -116,21 +116,22 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_number = len(class_group_ideals)
-    class_group_ideal_norms = [a.norm() for a in class_group_ideals]
-    norm_bound = bound*max(class_group_ideal_norms)
+    class_group_ideal_norms = [i.norm() for i in class_group_ideals]
+    norm_bound = bound * max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
     fund_unit_logs = list(map(log_embed, fundamental_units))
+    mat = column_matrix(fund_unit_logs)
 
-    test_matrix = column_matrix(fund_unit_logs)
-    #try:
-    #    test_matrix.change_ring(QQ)
-    #except ValueError:
-    #    raise ValueError('prec too low.')
+    test_matrix = mat
+    try:
+       test_matrix.change_ring(QQ)
+    except ValueError:
+       raise ValueError('prec too low.')
 
-    cut_fund_unit_logs = test_matrix.delete_rows([r])
+    cut_fund_unit_logs = mat.delete_rows([r])
     lll_fund_units = []
-    for c in cut_fund_unit_logs.columns():
     #for c in pari(cut_fund_unit_logs).qflll().python().columns():
+    for c in cut_fund_unit_logs.columns():
         new_unit = 1
         for i in range(r):
             print(fundamental_units[i])
@@ -172,7 +173,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     for v in range(r + 1):
         A_numbers.append(min([pr_ideal_gen_logs[y][v] for y in pr_ideal_gen_logs]))
 
-    aux_constant = (1/K_degree)*Reals(norm_bound).log()
+    aux_constant = (1/K_degree) * Reals(norm_bound).log()
 
     L_numbers = []
     for v in range(r1):
@@ -212,9 +213,9 @@ def points_of_bounded_height(K, dim, bound, prec=53):
                 y_log = pr_ideal_gen_logs[y]
                 g_log = unit_log + y_log
                 bool1 = all(g_log[i] <= aux_constant + D_numbers[i] for i in range(r1))
-                bool2 = all(g_log[j] <= 2*aux_constant + D_numbers[j] for j in range(r1, r + 1))
+                bool2 = all(g_log[j] <= 2 * aux_constant + D_numbers[j] for j in range(r1, r + 1))
                 if bool1 and bool2:
-                    g = unit*y
+                    g = unit * y
                     coordinate_list.append([g, g_log])
         if len(coordinate_list) > 0:
             coordinate_space[norm] = coordinate_list
@@ -227,13 +228,13 @@ def points_of_bounded_height(K, dim, bound, prec=53):
         a_coordinates = []
 
         for k in range(bound + 1):
-            norm = k*a_norm
+            norm = k * a_norm
             if norm in coordinate_space:
                 for pair in coordinate_space[norm]:
                     g, g_log = pair
                     if g in a:
                         bool1 = all(g_log[i] <= a_const + D_numbers[i] for i in range(r1))
-                        bool2 = all(g_log[j] <= 2*a_const + D_numbers[j] for j in range(r1, r + 1))
+                        bool2 = all(g_log[j] <= 2 * a_const + D_numbers[j] for j in range(r1, r + 1))
                         if bool1 and bool2:
                             a_coordinates.append(pair)
 

From fc09436787030034d389e1e458a5212d9a170c41 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 31 Aug 2022 14:02:10 +0800
Subject: [PATCH 18/42] restore commented line

---
 src/sage/schemes/projective/proj_bdd_height.py | 5 +----
 1 file changed, 1 insertion(+), 4 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index e20bf297962..b624038cd28 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -130,12 +130,9 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     cut_fund_unit_logs = mat.delete_rows([r])
     lll_fund_units = []
-    #for c in pari(cut_fund_unit_logs).qflll().python().columns():
-    for c in cut_fund_unit_logs.columns():
+    for c in pari(cut_fund_unit_logs).qflll().python():
         new_unit = 1
         for i in range(r):
-            print(fundamental_units[i])
-            print(c[i])
             new_unit *= fundamental_units[i]**c[i]
         lll_fund_units.append(new_unit)
     fundamental_units = lll_fund_units

From e150af850836823d0c733ab16d345818149b67a5 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 31 Aug 2022 21:29:28 +0800
Subject: [PATCH 19/42] examples

---
 .../schemes/projective/proj_bdd_height.py     | 26 +++++++++++++++++--
 .../schemes/projective/projective_space.py    |  7 +++++
 2 files changed, 31 insertions(+), 2 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index b624038cd28..01b0a7986e4 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -95,9 +95,29 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     OUTPUT:
 
     - an iterator of points of bounded height
+
+    EXAMPLES:
+
+        sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
+        sage: R.<x> = PolynomialRing(QQ)
+        sage: K.<a> = NumberField(x^3 + 5)
+        sage: list(points_of_bounded_height(K, 2, -3))
+        []
+        sage: list(points_of_bounded_height(K, 2, 2))
+        [(1 : 0 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 1 : 0),
+         (1 : 1 : 0), (-1 : 1 : 1), (1 : -1 : 1), (-1 : -1 : 1),
+         (1 : 0 : 0), (-1 : 1 : 0), (0 : 0 : 1), (1 : 1 : 1),
+         (0 : 1 : 1)]
+
+    ::
+
+        sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
+        sage: K.<a> = QuadraticField(3)
+        sage: list(points_of_bounded_height(K, 1, 1))
+        [(0 : 1), (1 : 0), (1 : 1), (-1 : 1)]
     """
     if bound < 1:
-        return []
+        return iter([])
 
     r1, r2 = K.signature()
     r = r1 + r2 - 1
@@ -249,4 +269,6 @@ def points_of_bounded_height(K, dim, bound, prec=53):
                         new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
                         points_in_class_a.add(PN(new_point))
         points_of_bdd_height += list(points_in_class_a)
-    return points_of_bdd_height
+
+    return iter(points_of_bdd_height)
+
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 93b8d8c7b8a..2adfaafe764 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1879,6 +1879,13 @@ def points_of_bounded_height(self, **kwds):
              (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (0 : 1),
              (1/3 : 1), (1/2 : 1), (2/3 : 1), (1 : 0),
              (1 : 1), (3/2 : 1), (2 : 1), (3 : 1)]
+
+        ::
+
+            sage: u = QQ['u'].0
+            sage: P.<x,y,z> = ProjectiveSpace(NumberField(u^2 - 2, 'v'), 2)
+            sage: len(list(P.points_of_bounded_height(bound=2)))
+            265
         """
         from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, points_of_bounded_height
 

From 1cc7b79098ebff1254eac1eb48e2892ef81e98e6 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 1 Sep 2022 13:03:12 +0800
Subject: [PATCH 20/42] more examples and rel degree

---
 .../schemes/projective/proj_bdd_height.py     |  6 +++-
 .../schemes/projective/projective_space.py    | 28 +++++++++++++++----
 2 files changed, 27 insertions(+), 7 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 01b0a7986e4..0a033719c6b 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -121,7 +121,11 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     r1, r2 = K.signature()
     r = r1 + r2 - 1
-    K_degree = K.degree()
+
+    if K.is_relative:
+        K_degree = K.relative_degree()
+    else:
+        K_degree = K.degree()
     K_embeddings = K.places(prec=prec)
     roots_of_unity = K.roots_of_unity()
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 2adfaafe764..3374298ceb8 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1863,7 +1863,7 @@ def points_of_bounded_height(self, **kwds):
 
         kwds:
 
-        - ``bound`` - a real number
+        - ``bound`` - an integer
 
         - ``precision`` - (default: 53) a positive integer
 
@@ -1885,7 +1885,25 @@ def points_of_bounded_height(self, **kwds):
             sage: u = QQ['u'].0
             sage: P.<x,y,z> = ProjectiveSpace(NumberField(u^2 - 2, 'v'), 2)
             sage: len(list(P.points_of_bounded_height(bound=2)))
-            265
+            13
+
+        ::
+
+            sage: CF.<a> = CyclotomicField(5)
+            sage: R.<x> = CF[]
+            sage: L.<l> = CF.extension(x^5 + 5)
+            sage: Q.<x,y> = ProjectiveSpace(L, 1)
+            sage: list(Q.points_of_bounded_height(bound=2))
+
+        ::
+
+            sage: R.<x> = QQ[]
+            sage: F.<a> = NumberField(x^4 - 8*x^2 + 3)
+            sage: P.<x,y,z> = ProjectiveSpace(F, 2)
+            sage: all([exp(p.global_height()) <= 1 for p in P.points_of_bounded_height(bound=1)])
+            True
+            sage: all([exp(p.global_height()) <= 3 for p in P.points_of_bounded_height(bound=3)])
+            True
         """
         from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, points_of_bounded_height
 
@@ -1901,16 +1919,14 @@ def points_of_bounded_height(self, **kwds):
             raise NotImplementedError("self must be projective space over a number field")
 
         bound = kwds.pop('bound')
-        B = bound**(self.base_ring().absolute_degree())  # convert to relative height
-
         prec = kwds.pop('precision', 53)
 
         dim = self.dimension_relative()
 
         if field_type:
-            return points_of_bounded_height(R, dim, B, prec)
+            return points_of_bounded_height(R, dim, bound, prec)
         else:
-            return QQ_points_of_bounded_height(dim, B)
+            return QQ_points_of_bounded_height(dim, bound)
 
     def subscheme_from_Chow_form(self, Ch, dim):
         r"""

From 26885489d3ebf9650467b7f0d2499c5d3d102b5f Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 1 Sep 2022 15:35:33 +0800
Subject: [PATCH 21/42] 34686: Support for relative number field

---
 src/sage/schemes/projective/proj_bdd_height.py  | 7 +++++--
 src/sage/schemes/projective/projective_space.py | 6 ++++--
 2 files changed, 9 insertions(+), 4 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 0a033719c6b..b8f8c472be1 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -122,7 +122,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     r1, r2 = K.signature()
     r = r1 + r2 - 1
 
-    if K.is_relative:
+    if K.is_relative():
         K_degree = K.relative_degree()
     else:
         K_degree = K.degree()
@@ -140,7 +140,10 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_number = len(class_group_ideals)
-    class_group_ideal_norms = [i.norm() for i in class_group_ideals]
+    if K.is_relative():
+        class_group_ideal_norms = [i.absolute_norm() for i in class_group_ideals]
+    else:
+        class_group_ideal_norms = [i.norm() for i in class_group_ideals]
     norm_bound = bound * max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
     fund_unit_logs = list(map(log_embed, fundamental_units))
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 3374298ceb8..7c559eef1af 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1889,11 +1889,13 @@ def points_of_bounded_height(self, **kwds):
 
         ::
 
-            sage: CF.<a> = CyclotomicField(5)
+            sage: CF.<a> = CyclotomicField(3)
             sage: R.<x> = CF[]
-            sage: L.<l> = CF.extension(x^5 + 5)
+            sage: L.<l> = CF.extension(x^3 + 2)
             sage: Q.<x,y> = ProjectiveSpace(L, 1)
             sage: list(Q.points_of_bounded_height(bound=2))
+            [(0 : 1), (a : 1), (a + 1 : 1), (-a - 1 : 1),
+             (-a : 1), (-1 : 1), (1 : 1), (1 : 0)]
 
         ::
 

From 83f385fc31a129c36389ac91cfd3c66d2d15c920 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sat, 3 Sep 2022 12:52:07 +0800
Subject: [PATCH 22/42] 32686: points_of_bounded_height for imaginary quadraitc
 field

---
 .../schemes/projective/proj_bdd_height.py     | 104 ++++++++++++------
 1 file changed, 71 insertions(+), 33 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index b8f8c472be1..b5b1cf865d9 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -18,6 +18,7 @@
 """
 
 import itertools
+
 from sage.schemes.projective.projective_space import ProjectiveSpace
 from sage.rings.rational_field import QQ
 from sage.rings.all import RealField
@@ -61,6 +62,7 @@ def QQ_points_of_bounded_height(dim, bound):
     """
     if bound < 1:
         return iter(set([]))
+
     PN = ProjectiveSpace(QQ, dim)
     unit_tuples = list(itertools.product([-1, 1], repeat=dim))
     points_of_bounded_height = set([])
@@ -71,6 +73,71 @@ def QQ_points_of_bounded_height(dim, bound):
                 for u in unit_tuples:
                     new_point = [a*b for a, b in zip(u, p)] + [p[dim]]
                     points_of_bounded_height.add(PN(new_point))
+
+    return iter(points_of_bounded_height)
+
+def IQ_points_of_bounded_height(K, dim, bound):
+    r"""
+    Return an iterator of the points in ``self`` of absolute height of
+    at most ``bound`` in the imaginary quadratic field ``K``.
+
+    INPUT:
+
+    - ``K`` -- a number field
+
+    - ``dim`` -- a positive interger
+
+    - ``bound`` -- a real number
+
+    OUTPUT:
+
+    - an iterator of points of bounded height
+    """
+    if bound < 1:
+        return iter(set([]))
+
+    PN = ProjectiveSpace(K, dim)
+    unit_tuples = list(itertools.product(K.roots_of_unity(), repeat=dim))
+    points_of_bounded_height = []
+    
+    class_group_ideals = [c.ideal() for c in K.class_group()]
+    class_group_ideal_norms = [i.norm() for i in class_group_ideals]
+    class_number = len(class_group_ideals)
+
+    possible_norm_set = set([])
+    for i in range(class_number):
+        for k in range(1, bound + 1):
+            possible_norm_set.add(k*class_group_ideal_norms[i])
+    
+    coordinate_space = dict()
+    coordinate_space[0] = [K(0)]
+    for m in possible_norm_set:
+        coordinate_space[m] = K.elements_of_norm(m)
+
+    for i in range(class_number):
+        a = class_group_ideals[i]
+        a_norm = class_group_ideal_norms[i]
+        a_norm_bound = bound * a_norm
+        a_coordinates = []
+
+        for m in coordinate_space:
+            if m <= a_norm_bound:
+                for x in coordinate_space[m]:
+                    if x in a:
+                        a_coordinates.append(x)
+
+        points_in_class_a = set([])
+        t = len(a_coordinates) - 1
+        increasing_tuples = itertools.combinations_with_replacement(range(t + 1), dim + 1)
+        for index_tuple in increasing_tuples:
+            point_coordinates = [a_coordinates[i] for i in index_tuple]
+            if a == K.ideal(point_coordinates):
+                for p in itertools.permutations(point_coordinates):
+                    for u in unit_tuples:
+                        new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
+                        points_in_class_a.add(PN(new_point))
+        points_of_bounded_height += list(points_in_class_a)
+
     return iter(points_of_bounded_height)
 
 def points_of_bounded_height(K, dim, bound, prec=53):
@@ -95,37 +162,13 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     OUTPUT:
 
     - an iterator of points of bounded height
-
-    EXAMPLES:
-
-        sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
-        sage: R.<x> = PolynomialRing(QQ)
-        sage: K.<a> = NumberField(x^3 + 5)
-        sage: list(points_of_bounded_height(K, 2, -3))
-        []
-        sage: list(points_of_bounded_height(K, 2, 2))
-        [(1 : 0 : 1), (-1 : 0 : 1), (0 : -1 : 1), (0 : 1 : 0),
-         (1 : 1 : 0), (-1 : 1 : 1), (1 : -1 : 1), (-1 : -1 : 1),
-         (1 : 0 : 0), (-1 : 1 : 0), (0 : 0 : 1), (1 : 1 : 1),
-         (0 : 1 : 1)]
-
-    ::
-
-        sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
-        sage: K.<a> = QuadraticField(3)
-        sage: list(points_of_bounded_height(K, 1, 1))
-        [(0 : 1), (1 : 0), (1 : 1), (-1 : 1)]
     """
     if bound < 1:
-        return iter([])
+        return []
 
     r1, r2 = K.signature()
     r = r1 + r2 - 1
-
-    if K.is_relative():
-        K_degree = K.relative_degree()
-    else:
-        K_degree = K.degree()
+    K_degree = K.degree()
     K_embeddings = K.places(prec=prec)
     roots_of_unity = K.roots_of_unity()
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
@@ -140,10 +183,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_number = len(class_group_ideals)
-    if K.is_relative():
-        class_group_ideal_norms = [i.absolute_norm() for i in class_group_ideals]
-    else:
-        class_group_ideal_norms = [i.norm() for i in class_group_ideals]
+    class_group_ideal_norms = [i.norm() for i in class_group_ideals]
     norm_bound = bound * max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
     fund_unit_logs = list(map(log_embed, fundamental_units))
@@ -276,6 +316,4 @@ def points_of_bounded_height(K, dim, bound, prec=53):
                         new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
                         points_in_class_a.add(PN(new_point))
         points_of_bdd_height += list(points_in_class_a)
-
-    return iter(points_of_bdd_height)
-
+    return points_of_bdd_height

From 2a67e06550a8d0b271c879194ee1d9b2010c0ab1 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sat, 3 Sep 2022 14:11:19 +0800
Subject: [PATCH 23/42] 32686: Support for imaginary quadratic field

---
 .../schemes/projective/proj_bdd_height.py     | 38 ++++++++++++++++---
 .../schemes/projective/projective_space.py    | 21 +++++++++-
 2 files changed, 53 insertions(+), 6 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index b5b1cf865d9..f4f7a831db7 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -92,9 +92,18 @@ def IQ_points_of_bounded_height(K, dim, bound):
     OUTPUT:
 
     - an iterator of points of bounded height
+
+    EXAMPLES:
+
+        sage: from sage.schemes.projective.proj_bdd_height import IQ_points_of_bounded_height
+        sage: CF.<a> = CyclotomicField(3)
+        sage: len(list(IQ_points_of_bounded_height(CF, 2, -1)))
+        0
+        sage: len(list(IQ_points_of_bounded_height(CF, 2, 1)))
+        57
     """
     if bound < 1:
-        return iter(set([]))
+        return iter([])
 
     PN = ProjectiveSpace(K, dim)
     unit_tuples = list(itertools.product(K.roots_of_unity(), repeat=dim))
@@ -162,13 +171,25 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     OUTPUT:
 
     - an iterator of points of bounded height
+
+    EXAMPLES:
+
+        sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
+        sage: K.<a> = NumberField(x^3 - 7)
+        sage: len(list(points_of_bounded_height(K, 2, 1)))
+        13
     """
     if bound < 1:
-        return []
+        return iter([])
 
     r1, r2 = K.signature()
     r = r1 + r2 - 1
-    K_degree = K.degree()
+
+    if K.is_relative():
+        K_degree = K.relative_degree()
+    else:
+        K_degree = K.degree()
+
     K_embeddings = K.places(prec=prec)
     roots_of_unity = K.roots_of_unity()
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
@@ -183,7 +204,12 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_number = len(class_group_ideals)
-    class_group_ideal_norms = [i.norm() for i in class_group_ideals]
+
+    if K.is_relative():
+        class_group_ideal_norms = [i.absolute_norm() for i in class_group_ideals]
+    else:
+        class_group_ideal_norms = [i.norm() for i in class_group_ideals]
+
     norm_bound = bound * max(class_group_ideal_norms)
     fundamental_units = UnitGroup(K).fundamental_units()
     fund_unit_logs = list(map(log_embed, fundamental_units))
@@ -316,4 +342,6 @@ def points_of_bounded_height(K, dim, bound, prec=53):
                         new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
                         points_in_class_a.add(PN(new_point))
         points_of_bdd_height += list(points_in_class_a)
-    return points_of_bdd_height
+
+    return iter(points_of_bdd_height)
+
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 7c559eef1af..03158bb1627 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1906,8 +1906,15 @@ def points_of_bounded_height(self, **kwds):
             True
             sage: all([exp(p.global_height()) <= 3 for p in P.points_of_bounded_height(bound=3)])
             True
+
+        ::
+
+            sage: K.<a> = CyclotomicField(3)
+            sage: P.<x,y,z> = ProjectiveSpace(K, 2)
+            sage: len(list(P.points_of_bounded_height(bound=1)))
+            57
         """
-        from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, points_of_bounded_height
+        from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, IQ_points_of_bounded_height, points_of_bounded_height
 
         R = self.base_ring()
 
@@ -1926,6 +1933,18 @@ def points_of_bounded_height(self, **kwds):
         dim = self.dimension_relative()
 
         if field_type:
+            # for imaginary quadratic field
+            r1, r2 = R.signature()
+            r = r1 + r2 - 1
+
+            if R.is_relative():
+                deg = R.relative_degree()
+            else:
+                deg = R.degree()
+
+            if deg == 2 and r == 0:
+                return IQ_points_of_bounded_height(R, dim, bound)
+
             return points_of_bounded_height(R, dim, bound, prec)
         else:
             return QQ_points_of_bounded_height(dim, bound)

From f4e99ee347bd6ac2aec981e281aab08873b38230 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sat, 3 Sep 2022 15:04:56 +0800
Subject: [PATCH 24/42] 32686: Raise degree to absolute degree

---
 .../schemes/projective/projective_space.py    | 32 ++++++++++++++-----
 1 file changed, 24 insertions(+), 8 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 03158bb1627..15d319b8f57 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1874,18 +1874,16 @@ def points_of_bounded_height(self, **kwds):
         EXAMPLES::
 
             sage: P.<x,y> = ProjectiveSpace(QQ, 1)
-            sage: sorted(list(P.points_of_bounded_height(bound=3)))
-            [(-3 : 1), (-2 : 1), (-3/2 : 1), (-1 : 1),
-             (-2/3 : 1), (-1/2 : 1), (-1/3 : 1), (0 : 1),
-             (1/3 : 1), (1/2 : 1), (2/3 : 1), (1 : 0),
-             (1 : 1), (3/2 : 1), (2 : 1), (3 : 1)]
+            sage: sorted(list(P.points_of_bounded_height(bound=2)))
+            [(-2 : 1), (-1 : 1), (-1/2 : 1), (0 : 1),
+             (1/2 : 1), (1 : 0), (1 : 1), (2 : 1)]
 
         ::
 
             sage: u = QQ['u'].0
             sage: P.<x,y,z> = ProjectiveSpace(NumberField(u^2 - 2, 'v'), 2)
             sage: len(list(P.points_of_bounded_height(bound=2)))
-            13
+            265
 
         ::
 
@@ -1893,9 +1891,9 @@ def points_of_bounded_height(self, **kwds):
             sage: R.<x> = CF[]
             sage: L.<l> = CF.extension(x^3 + 2)
             sage: Q.<x,y> = ProjectiveSpace(L, 1)
-            sage: list(Q.points_of_bounded_height(bound=2))
+            sage: sorted(list(Q.points_of_bounded_height(bound=1)))
             [(0 : 1), (a : 1), (a + 1 : 1), (-a - 1 : 1),
-             (-a : 1), (-1 : 1), (1 : 1), (1 : 0)]
+            (-a : 1), (-1 : 1), (1 : 1), (1 : 0)]
 
         ::
 
@@ -1913,6 +1911,22 @@ def points_of_bounded_height(self, **kwds):
             sage: P.<x,y,z> = ProjectiveSpace(K, 2)
             sage: len(list(P.points_of_bounded_height(bound=1)))
             57
+
+        ::
+
+            sage: u = QQ['u'].0
+            sage: K.<k> = NumberField(u^2 - 2)
+            sage: P.<x,y> = ProjectiveSpace(K, 1)
+            sage: len(list(P.points_of_bounded_height(bound=2)))
+            24
+
+        ::
+
+            sage: R.<x> = QQ[]
+            sage: K.<k> = NumberField(x^4 - 8*x^2 + 3)
+            sage: P.<x,y> = ProjectiveSpace(K, 1)
+            sage: len(list(P.points_of_bounded_height(bound=2)))
+            108
         """
         from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, IQ_points_of_bounded_height, points_of_bounded_height
 
@@ -1930,6 +1944,8 @@ def points_of_bounded_height(self, **kwds):
         bound = kwds.pop('bound')
         prec = kwds.pop('precision', 53)
 
+        bound = bound**R.absolute_degree()
+
         dim = self.dimension_relative()
 
         if field_type:

From 34f1439986bcd5acccb2809d0a14e62d2173c7b3 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 4 Sep 2022 12:21:22 +0800
Subject: [PATCH 25/42] 32686: Correct typo, add doc, and remove an unnecessary
 example

---
 src/sage/schemes/projective/proj_bdd_height.py  | 6 +++---
 src/sage/schemes/projective/projective_space.py | 3 +--
 2 files changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index f4f7a831db7..a3a213246dc 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -1,7 +1,7 @@
 r"""
 Points of bounded height in projective spaces
 
-This module defines sunctions to compute points of bounded height of a given
+This module defines functions to compute points of bounded height of a given
 number field with height less than a specified bound in projective spaces.
 
 Sage functions to list all elements of a given number field with height less
@@ -108,7 +108,7 @@ def IQ_points_of_bounded_height(K, dim, bound):
     PN = ProjectiveSpace(K, dim)
     unit_tuples = list(itertools.product(K.roots_of_unity(), repeat=dim))
     points_of_bounded_height = []
-    
+
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_group_ideal_norms = [i.norm() for i in class_group_ideals]
     class_number = len(class_group_ideals)
@@ -117,7 +117,7 @@ def IQ_points_of_bounded_height(K, dim, bound):
     for i in range(class_number):
         for k in range(1, bound + 1):
             possible_norm_set.add(k*class_group_ideal_norms[i])
-    
+
     coordinate_space = dict()
     coordinate_space[0] = [K(0)]
     for m in possible_norm_set:
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 15d319b8f57..acaf6683143 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1902,8 +1902,6 @@ def points_of_bounded_height(self, **kwds):
             sage: P.<x,y,z> = ProjectiveSpace(F, 2)
             sage: all([exp(p.global_height()) <= 1 for p in P.points_of_bounded_height(bound=1)])
             True
-            sage: all([exp(p.global_height()) <= 3 for p in P.points_of_bounded_height(bound=3)])
-            True
 
         ::
 
@@ -1944,6 +1942,7 @@ def points_of_bounded_height(self, **kwds):
         bound = kwds.pop('bound')
         prec = kwds.pop('precision', 53)
 
+        # Convert between absolute and relative height for calling Krumm's algorithm
         bound = bound**R.absolute_degree()
 
         dim = self.dimension_relative()

From 10c681c87c0de40448777a11dfc5c9e62774591c Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 4 Sep 2022 12:58:50 +0800
Subject: [PATCH 26/42] 32686: Correct some examples in
 hyperplane_transformation_matrix

---
 .../schemes/projective/projective_space.py    | 36 +++++++++----------
 1 file changed, 18 insertions(+), 18 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index acaf6683143..06da4cc06e0 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1547,8 +1547,8 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x)
             sage: plane2 = P.subscheme(y)
             sage: m = P.hyperplane_transformation_matrix(plane1, plane2); m
-            [-1 -1]
-            [ 1  0]
+            [0 1]
+            [1 0]
             sage: plane2(m*P((0,1)))
             (1 : 0)
 
@@ -1558,10 +1558,10 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + 2*y + z)
             sage: plane2 = P.subscheme(2*x + y + z)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [  -3    0    0    0]
-            [   9    6    0    0]
-            [-3/2   -3  3/2    0]
-            [-1/2   -1 -1/2    1]
+            [ 1/2    0    0    0]
+            [-1/2    1    0    0]
+            [   0    0  1/2    0]
+            [   0    0    0    1]
 
         ::
 
@@ -1569,8 +1569,8 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + y)
             sage: plane2 = P.subscheme(y)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [ 1  0]
-            [-1 -1]
+            [1 0]
+            [1 1]
 
         ::
 
@@ -1580,9 +1580,9 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane2 = P.subscheme(x + v*y + v*z)
             sage: m = P.hyperplane_transformation_matrix(plane1, plane2)
             sage: m
-            [ -6/7*v - 2/7             0             0]
-            [ 2/7*v + 10/7  -4/7*v + 8/7             0]
-            [ -4/7*v + 1/7 -10/7*v - 8/7             1]
+            [-4/21*v + 23/21 10/21*v + 16/21    8/21*v + 1/7]
+            [   8/21*v + 8/7   -8/21*v - 8/7   -8/21*v - 8/7]
+            [  4/21*v + 5/21  -2/21*v + 8/21               1]
 
         ::
 
@@ -1592,10 +1592,10 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(k*x + 2*k*y + z)
             sage: plane2 = P.subscheme(7*k*x + y + 9*z)
             sage: m = P.hyperplane_transformation_matrix(plane1, plane2); m
-            [   297/410*k + 279/410                      0                      0                      0]
-            [-3609/410*k + 4437/410 -1656/205*k + 2358/205                      0                      0]
-            [    511/410*k - 24/205     511/205*k - 48/205   -107/205*k + 327/410                      0]
-            [    83/410*k - 107/205     83/205*k - 214/205     107/205*k + 83/410                      1]
+            [     -4/65*k - 1/130                    0                    0                    0]
+            [    -25/52*k - 21/52   -139/130*k + 7/130  137/260*k - 121/260                    0]
+            [                   0                    0      -4/65*k - 1/130                    0]
+            [ 139/260*k + 123/260  139/130*k + 123/130 -123/260*k + 139/260                    1]
 
         ::
 
@@ -1627,9 +1627,9 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + 9*t*y + z)
             sage: plane2 = P.subscheme(x + z)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [       -1/9*t          -t^2             0]
-            [ -t^2 + 1/9*t             0             0]
-            [         1/81         1/9*t -1/9*t + 1/81]
+            [        -1/81        -1/9*t        -1/9*t]
+            [-1/9*t - 1/81   t^2 + 1/9*t  1/9*t + 1/81]
+            [       -1/9*t          -t^2         -1/81]
 
         TESTS::
 

From 2c4d407b911c42fdc26377ad9b418ecc4af40cfc Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sun, 4 Sep 2022 13:15:28 +0800
Subject: [PATCH 27/42] 32686: mention Trac ticket 11328 for insert_row

---
 src/sage/schemes/projective/proj_bdd_height.py | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index a3a213246dc..31b376f96bb 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -274,6 +274,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     T = column_matrix(fund_unit_logs).delete_rows([r]).change_ring(QQ)
 
+    # insert_row only takes integers, see https://trac.sagemath.org/ticket/11328
     M = ((-1)*matrix.identity(r)).insert_row(r, [Integer(1) for i in range(r)])
     M = M.transpose().insert_row(0, [Integer(0) for i in range(r + 1)]).transpose()
     M = M.change_ring(QQ)

From ac41c06e9e163d3e6ae1fd51b19f393289c779f3 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Fri, 9 Sep 2022 23:09:44 +0800
Subject: [PATCH 28/42] 32686: `bound` allows real numbers and `yield` instead
 of returning a list for points_of_bounded_height

---
 .../schemes/projective/proj_bdd_height.py     | 39 +++++++++----------
 1 file changed, 18 insertions(+), 21 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 31b376f96bb..1f1ca0e5e1e 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -19,6 +19,8 @@
 
 import itertools
 
+from math import floor
+
 from sage.schemes.projective.projective_space import ProjectiveSpace
 from sage.rings.rational_field import QQ
 from sage.rings.all import RealField
@@ -66,15 +68,15 @@ def QQ_points_of_bounded_height(dim, bound):
     PN = ProjectiveSpace(QQ, dim)
     unit_tuples = list(itertools.product([-1, 1], repeat=dim))
     points_of_bounded_height = set([])
-    increasing_tuples = itertools.combinations_with_replacement(range(bound + 1), dim + 1)
+    increasing_tuples = itertools.combinations_with_replacement(range(floor(bound + 1)), dim + 1)
     for t in increasing_tuples:
         if gcd(t) == 1:
             for p in itertools.permutations(t):
                 for u in unit_tuples:
-                    new_point = [a*b for a, b in zip(u, p)] + [p[dim]]
-                    points_of_bounded_height.add(PN(new_point))
-
-    return iter(points_of_bounded_height)
+                    PN_point = PN([a*b for a, b in zip(u, p)] + [p[dim]])
+                    if PN_point not in points_of_bounded_height:
+                        points_of_bounded_height.add(PN_point)
+                        yield PN_point
 
 def IQ_points_of_bounded_height(K, dim, bound):
     r"""
@@ -107,7 +109,6 @@ def IQ_points_of_bounded_height(K, dim, bound):
 
     PN = ProjectiveSpace(K, dim)
     unit_tuples = list(itertools.product(K.roots_of_unity(), repeat=dim))
-    points_of_bounded_height = []
 
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_group_ideal_norms = [i.norm() for i in class_group_ideals]
@@ -115,7 +116,7 @@ def IQ_points_of_bounded_height(K, dim, bound):
 
     possible_norm_set = set([])
     for i in range(class_number):
-        for k in range(1, bound + 1):
+        for k in range(1, ):
             possible_norm_set.add(k*class_group_ideal_norms[i])
 
     coordinate_space = dict()
@@ -142,12 +143,10 @@ def IQ_points_of_bounded_height(K, dim, bound):
             point_coordinates = [a_coordinates[i] for i in index_tuple]
             if a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
-                    for u in unit_tuples:
-                        new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
-                        points_in_class_a.add(PN(new_point))
-        points_of_bounded_height += list(points_in_class_a)
-
-    return iter(points_of_bounded_height)
+                    PN_point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
+                    if PN_point not in points_in_class_a:
+                        points_in_class_a.add(PN_point)
+                        yield PN_point
 
 def points_of_bounded_height(K, dim, bound, prec=53):
     r"""
@@ -233,7 +232,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     possible_norm_set = set([])
     for i in range(class_number):
-        for k in range(1, bound + 1):
+        for k in range(1, floor(bound + 1)):
             possible_norm_set.add(k*class_group_ideal_norms[i])
 
     principal_ideal_gens = dict()
@@ -318,7 +317,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
         a_const = (logB + log_a_norm)/K_degree
         a_coordinates = []
 
-        for k in range(bound + 1):
+        for k in range(floor(bound + 1)):
             norm = k * a_norm
             if norm in coordinate_space:
                 for pair in coordinate_space[norm]:
@@ -340,9 +339,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
             if log_arch_height <= log_arch_height_bound and a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        new_point = [i*j for i, j in zip(u, p)] + [p[dim]]
-                        points_in_class_a.add(PN(new_point))
-        points_of_bdd_height += list(points_in_class_a)
-
-    return iter(points_of_bdd_height)
-
+                        PN_point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
+                        if PN_point not in points_in_class_a:
+                            points_in_class_a.add(PN_point)
+                            yield PN_point

From 37f2d72672a9ce36772ab583fb6684ac0ea775d9 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Tue, 6 Sep 2022 10:33:15 +0800
Subject: [PATCH 29/42] 32686: Change ring when no common parent

---
 src/sage/schemes/projective/projective_space.py | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 06da4cc06e0..4d7c55e5fdd 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1616,8 +1616,8 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + y + z)
             sage: plane2 = P.subscheme(z)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [1 0 0]
-            [1 1 0]
+            [0 1 1]
+            [0 0 1]
             [1 1 1]
 
         ::
@@ -1698,7 +1698,7 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
                         source_points.append(self(point))
                         base_list = [list(s) for s in source_points]
                 elif len(source_points) == N + 1:
-                    Ms = matrix(base_list + [point])
+                    Ms = matrix(base_list + [point.change_ring(self.base_ring())])
                     if not any([m == 0 for m in Ms.minors(N + 1)]):
                         source_points.append(self(point))
                         break

From 31d0587e058db1053850c52c8dc76be59159a9a2 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sat, 10 Sep 2022 00:14:14 +0800
Subject: [PATCH 30/42] 32686: Change some example outputs

---
 src/sage/schemes/projective/proj_bdd_height.py  | 4 ++--
 src/sage/schemes/projective/projective_space.py | 4 ++--
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 1f1ca0e5e1e..fcd8557f459 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -51,8 +51,8 @@ def QQ_points_of_bounded_height(dim, bound):
     EXAMPLES:
 
         sage: from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height
-        sage: list(QQ_points_of_bounded_height(1, 1))
-        [(0 : 1), (1 : 0), (1 : 1), (-1 : 1)]
+        sage: sorted(list(QQ_points_of_bounded_height(1, 1)))
+        [(-1 : 1), (0 : 1), (1 : 0), (1 : 1)]
         sage: len(list(QQ_points_of_bounded_height(1, 5)))
         40
 
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 4d7c55e5fdd..806386b7151 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1892,8 +1892,8 @@ def points_of_bounded_height(self, **kwds):
             sage: L.<l> = CF.extension(x^3 + 2)
             sage: Q.<x,y> = ProjectiveSpace(L, 1)
             sage: sorted(list(Q.points_of_bounded_height(bound=1)))
-            [(0 : 1), (a : 1), (a + 1 : 1), (-a - 1 : 1),
-            (-a : 1), (-1 : 1), (1 : 1), (1 : 0)]
+            [(0 : 1), (1 : 0), (a + 1 : 1), (a : 1),
+             (-1 : 1), (-a - 1 : 1), (-a : 1), (1 : 1)]
 
         ::
 

From 8afc292bef6d28bba6bd2fc6d6347ef0f3565f7b Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sat, 10 Sep 2022 12:36:46 +0800
Subject: [PATCH 31/42] 32686: Debug `IQ_points_of_bounded_height`

---
 src/sage/schemes/projective/proj_bdd_height.py | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index fcd8557f459..8b0f04aaaa8 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -116,7 +116,7 @@ def IQ_points_of_bounded_height(K, dim, bound):
 
     possible_norm_set = set([])
     for i in range(class_number):
-        for k in range(1, ):
+        for k in range(1, floor(bound + 1)):
             possible_norm_set.add(k*class_group_ideal_norms[i])
 
     coordinate_space = dict()
@@ -143,10 +143,11 @@ def IQ_points_of_bounded_height(K, dim, bound):
             point_coordinates = [a_coordinates[i] for i in index_tuple]
             if a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
-                    PN_point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
-                    if PN_point not in points_in_class_a:
-                        points_in_class_a.add(PN_point)
-                        yield PN_point
+                    for u in unit_tuples:
+                        PN_point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
+                        if PN_point not in points_in_class_a:
+                            points_in_class_a.add(PN_point)
+                            yield PN_point
 
 def points_of_bounded_height(K, dim, bound, prec=53):
     r"""

From b3b05c8d3a852ab14fbb01a092002eacf225d0bd Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Sat, 10 Sep 2022 12:51:51 +0800
Subject: [PATCH 32/42] 32686: Change example outputs

---
 .../schemes/projective/projective_space.py    | 36 +++++++++----------
 1 file changed, 18 insertions(+), 18 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 806386b7151..9d60f748799 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1558,10 +1558,10 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + 2*y + z)
             sage: plane2 = P.subscheme(2*x + y + z)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [ 1/2    0    0    0]
-            [-1/2    1    0    0]
-            [   0    0  1/2    0]
-            [   0    0    0    1]
+            [1 0 0 0]
+            [0 4 0 0]
+            [0 0 2 0]
+            [0 0 0 1]
 
         ::
 
@@ -1569,8 +1569,8 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + y)
             sage: plane2 = P.subscheme(y)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [1 0]
-            [1 1]
+            [-1  0]
+            [ 1  1]
 
         ::
 
@@ -1580,9 +1580,9 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane2 = P.subscheme(x + v*y + v*z)
             sage: m = P.hyperplane_transformation_matrix(plane1, plane2)
             sage: m
-            [-4/21*v + 23/21 10/21*v + 16/21    8/21*v + 1/7]
-            [   8/21*v + 8/7   -8/21*v - 8/7   -8/21*v - 8/7]
-            [  4/21*v + 5/21  -2/21*v + 8/21               1]
+            [   v    0    0]
+            [   0 -2*v    0]
+            [   0    0    1]
 
         ::
 
@@ -1592,10 +1592,10 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(k*x + 2*k*y + z)
             sage: plane2 = P.subscheme(7*k*x + y + 9*z)
             sage: m = P.hyperplane_transformation_matrix(plane1, plane2); m
-            [     -4/65*k - 1/130                    0                    0                    0]
-            [    -25/52*k - 21/52   -139/130*k + 7/130  137/260*k - 121/260                    0]
-            [                   0                    0      -4/65*k - 1/130                    0]
-            [ 139/260*k + 123/260  139/130*k + 123/130 -123/260*k + 139/260                    1]
+            [   1    0    0    0]
+            [   0 14*k    0    0]
+            [   0    0  7/9    0]
+            [   0    0    0    1]
 
         ::
 
@@ -1616,8 +1616,8 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + y + z)
             sage: plane2 = P.subscheme(z)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [0 1 1]
-            [0 0 1]
+            [1 0 0]
+            [1 1 0]
             [1 1 1]
 
         ::
@@ -1627,9 +1627,9 @@ def hyperplane_transformation_matrix(self, plane_1, plane_2):
             sage: plane1 = P.subscheme(x + 9*t*y + z)
             sage: plane2 = P.subscheme(x + z)
             sage: P.hyperplane_transformation_matrix(plane1, plane2)
-            [        -1/81        -1/9*t        -1/9*t]
-            [-1/9*t - 1/81   t^2 + 1/9*t  1/9*t + 1/81]
-            [       -1/9*t          -t^2         -1/81]
+            [  1 9*t   0]
+            [  1   0   0]
+            [  0   0   1]
 
         TESTS::
 

From 5958fc6c65d31535b9580837ef4e614d3432a0cb Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 14 Sep 2022 11:35:59 +0800
Subject: [PATCH 33/42] 32686: Improve doc and return points in `self`

---
 .../schemes/projective/proj_bdd_height.py     | 37 +++++++++----------
 1 file changed, 17 insertions(+), 20 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 8b0f04aaaa8..421fe48381a 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -35,8 +35,8 @@
 
 def QQ_points_of_bounded_height(dim, bound):
     r"""
-    Return an iterator of the points in ``self`` of absolute height of
-    at most ``bound`` in the rational field.
+    Return an iterator of the points in ``self`` of absolute multiplicative
+    height of at most ``bound`` in the rational field.
 
     INPUT:
 
@@ -65,7 +65,6 @@ def QQ_points_of_bounded_height(dim, bound):
     if bound < 1:
         return iter(set([]))
 
-    PN = ProjectiveSpace(QQ, dim)
     unit_tuples = list(itertools.product([-1, 1], repeat=dim))
     points_of_bounded_height = set([])
     increasing_tuples = itertools.combinations_with_replacement(range(floor(bound + 1)), dim + 1)
@@ -73,15 +72,15 @@ def QQ_points_of_bounded_height(dim, bound):
         if gcd(t) == 1:
             for p in itertools.permutations(t):
                 for u in unit_tuples:
-                    PN_point = PN([a*b for a, b in zip(u, p)] + [p[dim]])
-                    if PN_point not in points_of_bounded_height:
-                        points_of_bounded_height.add(PN_point)
-                        yield PN_point
+                    point = self([a*b for a, b in zip(u, p)] + [p[dim]])
+                    if point not in points_of_bounded_height:
+                        points_of_bounded_height.add(point)
+                        yield point
 
 def IQ_points_of_bounded_height(K, dim, bound):
     r"""
-    Return an iterator of the points in ``self`` of absolute height of
-    at most ``bound`` in the imaginary quadratic field ``K``.
+    Return an iterator of the points in ``self`` of absolute multiplicative
+    height of at most ``bound`` in the imaginary quadratic field ``K``.
 
     INPUT:
 
@@ -107,7 +106,6 @@ def IQ_points_of_bounded_height(K, dim, bound):
     if bound < 1:
         return iter([])
 
-    PN = ProjectiveSpace(K, dim)
     unit_tuples = list(itertools.product(K.roots_of_unity(), repeat=dim))
 
     class_group_ideals = [c.ideal() for c in K.class_group()]
@@ -144,15 +142,15 @@ def IQ_points_of_bounded_height(K, dim, bound):
             if a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        PN_point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
-                        if PN_point not in points_in_class_a:
-                            points_in_class_a.add(PN_point)
-                            yield PN_point
+                        point = self([i*j for i, j in zip(u, p)] + [p[dim]])
+                        if point not in points_in_class_a:
+                            points_in_class_a.add(point)
+                            yield point
 
 def points_of_bounded_height(K, dim, bound, prec=53):
     r"""
     Return an iterator of the points in ``K`` with dimension ``dim`` of
-    absolute height of at most ``bound``.
+    absolute multiplicative height of at most ``bound``.
 
     ALGORITHM:
 
@@ -194,7 +192,6 @@ def points_of_bounded_height(K, dim, bound, prec=53):
     roots_of_unity = K.roots_of_unity()
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
 
-    PN = ProjectiveSpace(K, dim)
     log_embed = K.logarithmic_embedding()
 
     Reals = RealField(prec)
@@ -340,7 +337,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
             if log_arch_height <= log_arch_height_bound and a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        PN_point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
-                        if PN_point not in points_in_class_a:
-                            points_in_class_a.add(PN_point)
-                            yield PN_point
+                        point = self([i*j for i, j in zip(u, p)] + [p[dim]])
+                        if point not in points_in_class_a:
+                            points_in_class_a.add(point)
+                            yield point

From 1c80864f268e410aa7dfcc3598a9036579b0be7b Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 14 Sep 2022 11:52:40 +0800
Subject: [PATCH 34/42] 32686: Change `self` to `K` in non-rational
 points_of_bounded_height

---
 src/sage/schemes/projective/proj_bdd_height.py | 7 ++++---
 1 file changed, 4 insertions(+), 3 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 421fe48381a..b48625b58e3 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -65,6 +65,7 @@ def QQ_points_of_bounded_height(dim, bound):
     if bound < 1:
         return iter(set([]))
 
+    PN = ProjectiveSpace(QQ, dim)
     unit_tuples = list(itertools.product([-1, 1], repeat=dim))
     points_of_bounded_height = set([])
     increasing_tuples = itertools.combinations_with_replacement(range(floor(bound + 1)), dim + 1)
@@ -72,7 +73,7 @@ def QQ_points_of_bounded_height(dim, bound):
         if gcd(t) == 1:
             for p in itertools.permutations(t):
                 for u in unit_tuples:
-                    point = self([a*b for a, b in zip(u, p)] + [p[dim]])
+                    point = PN([a*b for a, b in zip(u, p)] + [p[dim]])
                     if point not in points_of_bounded_height:
                         points_of_bounded_height.add(point)
                         yield point
@@ -142,7 +143,7 @@ def IQ_points_of_bounded_height(K, dim, bound):
             if a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        point = self([i*j for i, j in zip(u, p)] + [p[dim]])
+                        point = K([i*j for i, j in zip(u, p)] + [p[dim]])
                         if point not in points_in_class_a:
                             points_in_class_a.add(point)
                             yield point
@@ -337,7 +338,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
             if log_arch_height <= log_arch_height_bound and a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        point = self([i*j for i, j in zip(u, p)] + [p[dim]])
+                        point = K([i*j for i, j in zip(u, p)] + [p[dim]])
                         if point not in points_in_class_a:
                             points_in_class_a.add(point)
                             yield point

From 5ec496e2a408c0d59f935252b32e4f91d6992740 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 14 Sep 2022 12:03:27 +0800
Subject: [PATCH 35/42] 32686: Add `PN` to args of non-rational points_bdd

---
 src/sage/schemes/projective/proj_bdd_height.py  | 16 +++++++++++-----
 src/sage/schemes/projective/projective_space.py |  4 ++--
 2 files changed, 13 insertions(+), 7 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index b48625b58e3..5f0bac1d35f 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -78,13 +78,15 @@ def QQ_points_of_bounded_height(dim, bound):
                         points_of_bounded_height.add(point)
                         yield point
 
-def IQ_points_of_bounded_height(K, dim, bound):
+def IQ_points_of_bounded_height(PN, K, dim, bound):
     r"""
     Return an iterator of the points in ``self`` of absolute multiplicative
     height of at most ``bound`` in the imaginary quadratic field ``K``.
 
     INPUT:
 
+    - ``PN`` -- a projective space
+
     - ``K`` -- a number field
 
     - ``dim`` -- a positive interger
@@ -99,7 +101,8 @@ def IQ_points_of_bounded_height(K, dim, bound):
 
         sage: from sage.schemes.projective.proj_bdd_height import IQ_points_of_bounded_height
         sage: CF.<a> = CyclotomicField(3)
-        sage: len(list(IQ_points_of_bounded_height(CF, 2, -1)))
+        sage: P.<x,y,z> = ProjectiveSpace(CF, 2)
+        sage: len(list(IQ_points_of_bounded_height(P, CF, 2, -1)))
         0
         sage: len(list(IQ_points_of_bounded_height(CF, 2, 1)))
         57
@@ -143,12 +146,12 @@ def IQ_points_of_bounded_height(K, dim, bound):
             if a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        point = K([i*j for i, j in zip(u, p)] + [p[dim]])
+                        point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
                         if point not in points_in_class_a:
                             points_in_class_a.add(point)
                             yield point
 
-def points_of_bounded_height(K, dim, bound, prec=53):
+def points_of_bounded_height(PN, K, dim, bound, prec=53):
     r"""
     Return an iterator of the points in ``K`` with dimension ``dim`` of
     absolute multiplicative height of at most ``bound``.
@@ -159,6 +162,8 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
     INPUT:
 
+    - ``PN`` -- a projective space
+
     - ``K`` -- a number field
 
     - ``dim`` -- a positive interger
@@ -175,6 +180,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
 
         sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
         sage: K.<a> = NumberField(x^3 - 7)
+        sage: P.<x,y,z> = ProjectiveSpace(K, 2)
         sage: len(list(points_of_bounded_height(K, 2, 1)))
         13
     """
@@ -338,7 +344,7 @@ def points_of_bounded_height(K, dim, bound, prec=53):
             if log_arch_height <= log_arch_height_bound and a == K.ideal(point_coordinates):
                 for p in itertools.permutations(point_coordinates):
                     for u in unit_tuples:
-                        point = K([i*j for i, j in zip(u, p)] + [p[dim]])
+                        point = PN([i*j for i, j in zip(u, p)] + [p[dim]])
                         if point not in points_in_class_a:
                             points_in_class_a.add(point)
                             yield point
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 9d60f748799..b882767edbe 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1958,9 +1958,9 @@ def points_of_bounded_height(self, **kwds):
                 deg = R.degree()
 
             if deg == 2 and r == 0:
-                return IQ_points_of_bounded_height(R, dim, bound)
+                return IQ_points_of_bounded_height(self, R, dim, bound)
 
-            return points_of_bounded_height(R, dim, bound, prec)
+            return points_of_bounded_height(self, R, dim, bound, prec)
         else:
             return QQ_points_of_bounded_height(dim, bound)
 

From 2adb101bc163c52a029181a7df3ad91acc0c41ed Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 14 Sep 2022 12:10:26 +0800
Subject: [PATCH 36/42] 32686: Correct tests

---
 src/sage/schemes/projective/proj_bdd_height.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 5f0bac1d35f..0cc57507158 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -104,7 +104,7 @@ def IQ_points_of_bounded_height(PN, K, dim, bound):
         sage: P.<x,y,z> = ProjectiveSpace(CF, 2)
         sage: len(list(IQ_points_of_bounded_height(P, CF, 2, -1)))
         0
-        sage: len(list(IQ_points_of_bounded_height(CF, 2, 1)))
+        sage: len(list(IQ_points_of_bounded_height(P, CF, 2, 1)))
         57
     """
     if bound < 1:
@@ -181,7 +181,7 @@ def points_of_bounded_height(PN, K, dim, bound, prec=53):
         sage: from sage.schemes.projective.proj_bdd_height import points_of_bounded_height
         sage: K.<a> = NumberField(x^3 - 7)
         sage: P.<x,y,z> = ProjectiveSpace(K, 2)
-        sage: len(list(points_of_bounded_height(K, 2, 1)))
+        sage: len(list(points_of_bounded_height(P, K, 2, 1)))
         13
     """
     if bound < 1:

From a23735269b76390cc7dde97c500d0d9974510ff7 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Wed, 14 Sep 2022 12:17:04 +0800
Subject: [PATCH 37/42] 32686: Improve doc

---
 src/sage/schemes/projective/projective_space.py | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index b882767edbe..c68157714eb 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1852,8 +1852,8 @@ def _morphism(self, *args, **kwds):
 
     def points_of_bounded_height(self, **kwds):
         r"""
-        Return an iterator of the points in ``self`` of absolute height of
-        at most the given bound.
+        Return an iterator of the points in ``self`` of absolute multiplicative
+        height of at most the given bound.
 
         ALGORITHM:
 

From 00af5c7d5f07ce95060549ef74ac1a2439f1cc13 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 22 Sep 2022 14:37:48 +0800
Subject: [PATCH 38/42] 32686: Correct doc and add an example

---
 src/sage/schemes/projective/projective_space.py | 10 +++++++++-
 1 file changed, 9 insertions(+), 1 deletion(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index c68157714eb..7facb20875f 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1863,7 +1863,7 @@ def points_of_bounded_height(self, **kwds):
 
         kwds:
 
-        - ``bound`` - an integer
+        - ``bound`` - a real number
 
         - ``precision`` - (default: 53) a positive integer
 
@@ -1925,6 +1925,14 @@ def points_of_bounded_height(self, **kwds):
             sage: P.<x,y> = ProjectiveSpace(K, 1)
             sage: len(list(P.points_of_bounded_height(bound=2)))
             108
+
+        ::
+
+            R.<x> = QQ[]
+            K.<v> = NumberField(x^5 + x^3 + 1)
+            P.<x,y,z> = ProjectiveSpace(K, 2)
+            L = P.points_of_bounded_height(bound=1.2)
+            0
         """
         from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, IQ_points_of_bounded_height, points_of_bounded_height
 

From 98a467602c4f4959cb847963ad08ade3334841b9 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 22 Sep 2022 15:16:57 +0800
Subject: [PATCH 39/42] 32686: Correct example

---
 src/sage/schemes/projective/projective_space.py | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 7facb20875f..117e071a67b 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -1928,11 +1928,12 @@ def points_of_bounded_height(self, **kwds):
 
         ::
 
-            R.<x> = QQ[]
-            K.<v> = NumberField(x^5 + x^3 + 1)
-            P.<x,y,z> = ProjectiveSpace(K, 2)
-            L = P.points_of_bounded_height(bound=1.2)
-            0
+            sage: R.<x> = QQ[]
+            sage: K.<v> = NumberField(x^5 + x^3 + 1)
+            sage: P.<x,y,z> = ProjectiveSpace(K, 2)
+            sage: L = P.points_of_bounded_height(bound=1.2)
+            sage: len(list(L))
+            109
         """
         from sage.schemes.projective.proj_bdd_height import QQ_points_of_bounded_height, IQ_points_of_bounded_height, points_of_bounded_height
 

From c4aab9f9dc11d787945fad21be16beee4b025456 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Thu, 6 Oct 2022 12:14:43 +0800
Subject: [PATCH 40/42] 32686: Remove unused imports and variables

---
 src/sage/schemes/projective/proj_bdd_height.py  | 3 ---
 src/sage/schemes/projective/projective_space.py | 1 -
 2 files changed, 4 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index 0cc57507158..c36ca173fd9 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -195,7 +195,6 @@ def points_of_bounded_height(PN, K, dim, bound, prec=53):
     else:
         K_degree = K.degree()
 
-    K_embeddings = K.places(prec=prec)
     roots_of_unity = K.roots_of_unity()
     unit_tuples = list(itertools.product(roots_of_unity, repeat=dim))
 
@@ -204,8 +203,6 @@ def points_of_bounded_height(PN, K, dim, bound, prec=53):
     Reals = RealField(prec)
     logB = Reals(bound).log()
 
-    points_of_bdd_height = []
-
     class_group_ideals = [c.ideal() for c in K.class_group()]
     class_number = len(class_group_ideals)
 
diff --git a/src/sage/schemes/projective/projective_space.py b/src/sage/schemes/projective/projective_space.py
index 117e071a67b..e4d100caa1e 100644
--- a/src/sage/schemes/projective/projective_space.py
+++ b/src/sage/schemes/projective/projective_space.py
@@ -80,7 +80,6 @@
 # ****************************************************************************
 
 from sage.arith.misc import gcd, binomial
-from sage.arith.srange import srange
 
 from sage.rings.finite_rings.finite_field_constructor import is_FiniteField
 from sage.rings.integer import Integer

From 10df9a648550ddf79315e9a6c9843eed01415d08 Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Fri, 7 Oct 2022 14:41:46 +0800
Subject: [PATCH 41/42] 32686: Correct example outputs in various files

---
 .../arithmetic_dynamics/projective_ds.py      | 26 ++++++++++---------
 .../schemes/projective/projective_homset.py   |  2 +-
 .../projective/projective_rational_point.py   |  3 +--
 3 files changed, 16 insertions(+), 15 deletions(-)

diff --git a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
index cdfc11a9e52..f6d811f972a 100644
--- a/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
+++ b/src/sage/dynamics/arithmetic_dynamics/projective_ds.py
@@ -3072,13 +3072,18 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             sage: g = f.affine_preperiodic_model(0, 1); g
             Dynamical System of Projective Space of dimension 2 over Rational Field
               Defn: Defined on coordinates by sending (x : y : z) to
-                    (-x^2 : 2*x^2 + 2*x*y + y^2 : 2*x^2 + 2*x*y + 2*y^2 - 2*y*z + z^2)
+                    (-x^2 : -2*x^2 + 2*x*y - y^2 : 2*x^2 - 2*x*y + 2*y^2 + 2*y*z + z^2)
 
         We can check that ``g`` has affine fixed points::
 
             sage: g.periodic_points(1)
-            [(-1 : 1 : 1), (-1/2 : 1/2 : 1), (-1/2 : 1 : 1), (-1/3 : 2/3 : 1), (0 : 0 : 1),
-            (0 : 1/2 : 1), (0 : 1 : 1)]
+            [(-1 : -1 : 1),
+            (-1/2 : -1 : 1),
+            (-1/2 : -1/2 : 1),
+            (-1/3 : -2/3 : 1),
+            (0 : -1 : 1),
+            (0 : -1/2 : 1),
+            (0 : 0 : 1)]
 
         ::
 
@@ -3087,8 +3092,8 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             sage: f.affine_preperiodic_model(0, 1)
             Dynamical System of Projective Space of dimension 2 over Finite Field in z2 of size 3^2
                   Defn: Defined on coordinates by sending (x : y : z) to
-                        ((z2 + 1)*x^2 : (z2 + 1)*x^2 + (z2 + 1)*x*y + (-z2 - 1)*y^2 :
-                        (z2 - 1)*x^2 + (z2 - 1)*x*y - y^2 + (-z2)*y*z + z^2)
+                        ((-z2)*x^2 : z2*x^2 + (-z2)*x*y + (-z2)*y^2 :
+                        (-z2)*x^2 + z2*x*y + (z2 + 1)*y^2 - y*z + z^2)
 
         ::
 
@@ -3099,9 +3104,8 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             Dynamical System of Projective Space of dimension 2 over
             Univariate Polynomial Ring in c over Finite Field of size 3
               Defn: Defined on coordinates by sending (x : y : z) to
-                    ((2*c^4 + c^3)*x^2 : (2*c^4 + c^3)*x^2 + (2*c^4 + c^3)*x*y + (c^4 + 2*c^3)*y^2 :
-                    c^3*x^2 + c^3*x*y + (2*c^3 + 2*c^2)*y^2 + (c^3 + 2*c^2)*y*z + (2*c^4 + 2*c^3 +
-                    2*c^2)*z^2)
+                    (2*c^3*x^2 : c^3*x^2 + 2*c^3*x*y + 2*c^3*y^2 :
+                    2*c^3*x^2 + c^3*x*y + (c^3 + c^2)*y^2 + 2*c^2*y*z + c^2*z^2)
 
         ::
 
@@ -3112,8 +3116,7 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             Dynamical System of Projective Space of dimension 2
             over Cyclotomic Field of order 3 and degree 2
               Defn: Defined on coordinates by sending (x : y : z) to
-                    (x^2 + y^2 + (-k + 2)*x*z - 2*y*z + (-k + 3)*z^2 :
-                    -2*x^2 + (k - 4)*x*z + (k - 3)*z^2 : -x^2 + (k - 2)*x*z + (k - 2)*z^2)
+                    (-y^2 : x^2 : x^2 + (-k)*x*z + z^2)
 
         ::
 
@@ -3132,8 +3135,7 @@ def affine_preperiodic_model(self, m, n, return_conjugation=False):
             Dynamical System of Closed subscheme of Projective Space of dimension 2 over Rational Field defined by:
               2*y - z
               Defn: Defined on coordinates by sending (x : y : z) to
-                    (2*x^2 + y^2 + 4*x*z - 2*y*z + 4*z^2 : -x^2 - y^2 - 2*x*z + 2*y*z - 3*z^2 :
-                    -x^2 - 2*x*z - 2*z^2)
+                    (-x^2 - y^2 : y^2 : x^2 + z^2)
 
         TESTS::
 
diff --git a/src/sage/schemes/projective/projective_homset.py b/src/sage/schemes/projective/projective_homset.py
index 0f7b6abce52..f9a8e69bd4b 100644
--- a/src/sage/schemes/projective/projective_homset.py
+++ b/src/sage/schemes/projective/projective_homset.py
@@ -124,7 +124,7 @@ def points(self, **kwds):
             sage: K.<v> = NumberField(u^2 + 3)
             sage: P.<x,y,z> = ProjectiveSpace(K,2)
             sage: len(P(K).points(bound=1.8))
-            381
+            309
 
         ::
 
diff --git a/src/sage/schemes/projective/projective_rational_point.py b/src/sage/schemes/projective/projective_rational_point.py
index 93789be8d65..cb12e30c873 100644
--- a/src/sage/schemes/projective/projective_rational_point.py
+++ b/src/sage/schemes/projective/projective_rational_point.py
@@ -191,8 +191,7 @@ def enum_projective_number_field(X, **kwds):
         sage: P.<x,y,z> = ProjectiveSpace(K, 2)
         sage: X = P.subscheme([x - y])
         sage: enum_projective_number_field(X(K), bound=RR(5^(1/3)), prec=2^10)
-        [(0 : 0 : 1), (-1 : -1 : 1), (1 : 1 : 1), (-1/5*v^2 : -1/5*v^2 : 1), (-v : -v : 1),
-        (1/5*v^2 : 1/5*v^2 : 1), (v : v : 1), (1 : 1 : 0)]
+        [(0 : 0 : 1), (1 : 1 : 0), (-1 : -1 : 1), (1 : 1 : 1)]
 
     ::
 

From c053c2ac5367b08f553d6dd3d4fa86dec14bc09d Mon Sep 17 00:00:00 2001
From: Jing Guo <dev.guoj@gmail.com>
Date: Fri, 7 Oct 2022 18:42:27 +0800
Subject: [PATCH 42/42] 32686: Format according to lint result

---
 src/sage/schemes/projective/proj_bdd_height.py | 6 ++++--
 1 file changed, 4 insertions(+), 2 deletions(-)

diff --git a/src/sage/schemes/projective/proj_bdd_height.py b/src/sage/schemes/projective/proj_bdd_height.py
index c36ca173fd9..33d5ec1d3bc 100644
--- a/src/sage/schemes/projective/proj_bdd_height.py
+++ b/src/sage/schemes/projective/proj_bdd_height.py
@@ -78,6 +78,7 @@ def QQ_points_of_bounded_height(dim, bound):
                         points_of_bounded_height.add(point)
                         yield point
 
+
 def IQ_points_of_bounded_height(PN, K, dim, bound):
     r"""
     Return an iterator of the points in ``self`` of absolute multiplicative
@@ -151,6 +152,7 @@ def IQ_points_of_bounded_height(PN, K, dim, bound):
                             points_in_class_a.add(point)
                             yield point
 
+
 def points_of_bounded_height(PN, K, dim, bound, prec=53):
     r"""
     Return an iterator of the points in ``K`` with dimension ``dim`` of
@@ -218,9 +220,9 @@ def points_of_bounded_height(PN, K, dim, bound, prec=53):
 
     test_matrix = mat
     try:
-       test_matrix.change_ring(QQ)
+        test_matrix.change_ring(QQ)
     except ValueError:
-       raise ValueError('prec too low.')
+        raise ValueError('prec too low.')
 
     cut_fund_unit_logs = mat.delete_rows([r])
     lll_fund_units = []