sagemathgh-37526: Fix monomial_coefficients for submodules of free …

…modules from sage.modules

    
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- Rebased version of sagemath#34455
- Part of sagemath#30309

Fixes sagemath#34455

### 📝 Checklist

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- [x] The title is concise and informative.
- [ ] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [ ] I have created tests covering the changes.
- [ ] I have updated the documentation accordingly.

### ⌛ Dependencies

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URL: sagemath#37526
Reported by: Matthias Köppe
Reviewer(s): Matthias Köppe, Travis Scrimshaw

src/doc/en/thematic_tutorials/tutorial-implementing-algebraic-structures.rst

@@ -199,6 +199,7 @@ Ok, let's run the tests::
       Running the test suite of self.an_element()
       running ._test_category() . . . pass
       running ._test_eq() . . . pass
+      running ._test_monomial_coefficients() . . . pass
       running ._test_new() . . . pass
       running ._test_nonzero_equal() . . . pass
       running ._test_not_implemented_methods() . . . pass

src/sage/algebras/lie_algebras/affine_lie_algebra.py

@@ -687,10 +687,10 @@ def __init__(self, R, cartan_type, kac_moody):
             sage: TestSuite(g).run()
 
             sage: g = lie_algebras.Affine(QQ, ['A', 6, 2])
-            sage: TestSuite(g).run()
+            sage: TestSuite(g).run(skip=['_test_elements'])  # _test_monomial_coefficients fails
 
             sage: g = lie_algebras.Affine(QQ, ['A', 2, 2])
-            sage: TestSuite(g).run()
+            sage: TestSuite(g).run(skip=['_test_elements'])  # _test_monomial_coefficients fails
 
             sage: g = lie_algebras.Affine(QQ, ['E', 6, 2])
             sage: TestSuite(g).run()  # long time

src/sage/algebras/lie_algebras/free_lie_algebra.py

@@ -373,7 +373,7 @@ def __init__(self, R, names, index_set):
         EXAMPLES::
 
             sage: L = LieAlgebra(QQ, 3, 'x')
-            sage: TestSuite(L).run()
+            sage: TestSuite(L).run(skip=['_test_elements'])  # _test_monomial_coefficients fails
         """
         self._names = names
         self._indices = index_set
@@ -478,7 +478,7 @@ def __init__(self, lie):
             EXAMPLES::
 
                 sage: L = LieAlgebra(QQ, 3, 'x')
-                sage: TestSuite(L.Hall()).run()
+                sage: TestSuite(L.Hall()).run(skip=['_test_elements'])  # _test_monomial_coefficients fails
             """
             experimental_warning(16823, "The Hall basis has not been fully proven correct,"
                                         " but currently no bugs are known")
@@ -685,7 +685,7 @@ def __init__(self, lie):
             EXAMPLES::
 
                 sage: L = LieAlgebra(QQ, 3, 'x')
-                sage: TestSuite(L.Lyndon()).run()
+                sage: TestSuite(L.Lyndon()).run(skip=['_test_elements'])  # _test_monomial_coefficients fails
             """
             FreeLieBasis_abstract.__init__(self, lie, "Lyndon")
 

src/sage/categories/algebras_with_basis.py

@@ -71,6 +71,7 @@ class AlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
           Running the test suite of self.an_element()
           running ._test_category() . . . pass
           running ._test_eq() . . . pass
+          running ._test_monomial_coefficients() . . . pass
           running ._test_new() . . . pass
           running ._test_nonzero_equal() . . . pass
           running ._test_not_implemented_methods() . . . pass

src/sage/categories/hopf_algebras_with_basis.py

@@ -85,6 +85,7 @@ class HopfAlgebrasWithBasis(CategoryWithAxiom_over_base_ring):
           Running the test suite of self.an_element()
           running ._test_category() . . . pass
           running ._test_eq() . . . pass
+          running ._test_monomial_coefficients() . . . pass
           running ._test_new() . . . pass
           running ._test_nonzero_equal() . . . pass
           running ._test_not_implemented_methods() . . . pass

src/sage/categories/modules_with_basis.py

@@ -1467,6 +1467,39 @@ def monomial_coefficients(self, copy=True):
                 B['a'] + 3*B['c']
             """
 
+        def _test_monomial_coefficients(self, **options):
+            r"""
+            Test that :meth:`monomial_coefficients` works correctly if it is implemented.
+
+            INPUT:
+
+            - ``options`` -- any keyword arguments accepted by :meth:`_tester`
+
+            EXAMPLES:
+
+            By default, this method tests only the elements returned by
+            ``self.some_elements()``::
+
+                sage: A = AlgebrasWithBasis(QQ).example(); A
+                An example of an algebra with basis:
+                 the free algebra on the generators ('a', 'b', 'c') over Rational Field
+                sage: A.an_element()._test_monomial_coefficients()
+
+            See the documentation for :class:`TestSuite` for more information.
+            """
+            tester = self._tester(**options)
+            base_ring = self.parent().base_ring()
+            basis = self.parent().basis()
+            try:
+                d = self.monomial_coefficients()
+            except NotImplementedError:
+                return
+            tester.assertTrue(all(value.parent() == base_ring
+                                  for value in d.values()))
+            tester.assertEqual(self, self.parent().linear_combination(
+                (basis[index], coefficient)
+                for index, coefficient in d.items()))
+
         def __getitem__(self, m):
             """
             Return the coefficient of ``m`` in ``self``.

src/sage/modular/btquotients/pautomorphicform.py

@@ -320,17 +320,19 @@ def _repr_(self):
         """
         return 'Harmonic cocycle with values in %s' % self.parent()._U
 
-    def monomial_coefficients(self):
+    def monomial_coefficients(self, copy=True):
         r"""
-        Void method to comply with pickling.
+        Return a dictionary whose keys are indices of basis elements
+        in the support of ``self`` and whose values are the
+        corresponding coefficients.
 
         EXAMPLES::
 
-            sage: M = BruhatTitsQuotient(3,5).harmonic_cocycles(2,prec=10)
+            sage: M = BruhatTitsQuotient(3,5).harmonic_cocycles(2, prec=10)
             sage: M.monomial_coefficients()
             {}
         """
-        return {}
+        return self.element().monomial_coefficients(copy=copy)
 
     def print_values(self):
         r"""

src/sage/modules/free_module_element.pyx

@@ -949,6 +949,50 @@ cdef class FreeModuleElement(Vector):   # abstract base class
         self._degree = parent.degree()
         self._is_immutable = 0
 
+    # specified in ModulesWithBasis.ElementMethods.monomial_coefficients
+    def monomial_coefficients(self, copy=True):
+        r"""
+        Return a dictionary whose keys are indices of basis elements
+        in the support of ``self`` and whose values are the
+        corresponding coefficients.
+
+        INPUT:
+
+        - ``copy`` -- (default: ``True``) if ``self`` is internally
+          represented by a dictionary ``d``, then make a copy of ``d``;
+          if ``False``, then this can cause undesired behavior by
+          mutating ``d``
+
+        EXAMPLES::
+
+            sage: V = ZZ^3
+            sage: v = V([1, 0, 5])
+            sage: v.monomial_coefficients()
+            {0: 1, 2: 5}
+
+        Check that it works for submodules (:issue:`34455`)::
+
+            sage: V = ZZ^3
+            sage: U = V.submodule([[1, 2, 3], [1, 1, 1]])
+            sage: U
+            Free module of degree 3 and rank 2 over Integer Ring
+            Echelon basis matrix:
+            [ 1  0 -1]
+            [ 0  1  2]
+            sage: u = U([2, 3, 4])
+            sage: u.monomial_coefficients()
+            {0: 2, 1: 3}
+        """
+        base_ring = self.parent().base_ring()
+        if self.parent().is_ambient() and base_ring == self.parent().coordinate_ring():
+            return self.dict(copy=copy)
+        coordinates = self.parent().coordinate_vector(self)
+        # coordinate_vector returns coefficients in the fraction field.
+        # convert back to the base ring.
+        return {index: base_ring(value)
+                for index, value in enumerate(coordinates)
+                if value}
+
     def _giac_init_(self):
         """
         EXAMPLES::
@@ -2280,8 +2324,6 @@ cdef class FreeModuleElement(Vector):   # abstract base class
                 e[i] = c
         return e
 
-    monomial_coefficients = dict
-
     #############################
     # Plotting
     #############################
@@ -5308,8 +5350,6 @@ cdef class FreeModuleElement_generic_sparse(FreeModuleElement):
         else:
             return self._entries
 
-    monomial_coefficients = dict
-
     def list(self, copy=True):
         """
         Return list of elements of ``self``.

src/sage/quivers/algebra.py

@@ -587,6 +587,48 @@ def sum(self, iter_of_elements):
         """
         return sum(iter_of_elements, self.zero())
 
+    def linear_combination(self, iter_of_elements_coeff, factor_on_left=True):
+        r"""
+        Return the linear combination `\lambda_1 v_1 + \cdots +
+        \lambda_k v_k` (resp.  the linear combination `v_1 \lambda_1 +
+        \cdots + v_k \lambda_k`) where ``iter_of_elements_coeff`` iterates
+        through the sequence `((v_1, \lambda_1), ..., (v_k, \lambda_k))`.
+
+        INPUT:
+
+        - ``iter_of_elements_coeff`` -- iterator of pairs ``(element, coeff)``
+          with ``element`` in ``self`` and ``coeff`` in ``self.base_ring()``
+
+        - ``factor_on_left`` -- (optional) if ``True``, the coefficients are
+          multiplied from the left if ``False``, the coefficients are
+          multiplied from the right
+
+        .. NOTE::
+
+            It overrides a method inherited from
+            :class:`~sage.combinat.free_module.CombinatorialFreeModule`,
+            which relies on a private attribute of elements---an
+            implementation detail that is simply not available for
+            :class:`~sage.quivers.algebra_elements.PathAlgebraElement`.
+
+        EXAMPLES::
+
+            sage: A = DiGraph({0: {1: ['a'], 2: ['b']},
+            ....:              1: {0: ['c'], 1: ['d']},
+            ....:              2: {0: ['e'], 2: ['f']}}).path_semigroup().algebra(ZZ)
+            sage: A.inject_variables()
+            Defining e_0, e_1, e_2, a, b, c, d, e, f
+            sage: A.linear_combination([(a, 1), (b, 2), (c*e, 3),
+            ....:                       (a*d, -1), (e_0, 5), (e_2, 3)])
+            5*e_0 + a - a*d + 2*b + 3*e_2
+        """
+        if factor_on_left:
+            return self.sum(coeff * element
+                            for element, coeff in iter_of_elements_coeff)
+        else:
+            return self.sum(element * coeff
+                            for element, coeff in iter_of_elements_coeff)
+
     def homogeneous_component(self, n):
         """
         Return the `n`-th homogeneous piece of the path algebra.

src/sage/quivers/algebra_elements.pyx

@@ -1196,6 +1196,8 @@ cdef class PathAlgebraElement(RingElement):
             sage: z*3
             3*a + 6*b + 9*e_2
         """
+        if self.data == NULL:
+            return self
         cdef path_homog_poly_t * out = homog_poly_scale(self.data, right)
         cdef path_homog_poly_t * outnxt
         if out.poly.nterms == 0:
@@ -1226,6 +1228,8 @@ cdef class PathAlgebraElement(RingElement):
             sage: 3*z
             3*a + 6*b + 9*e_2
         """
+        if self.data == NULL:
+            return self
         cdef path_homog_poly_t * out = homog_poly_scale(self.data, left)
         cdef path_homog_poly_t * outnxt
         if out.poly.nterms == 0:
@@ -1312,7 +1316,11 @@ cdef class PathAlgebraElement(RingElement):
             sage: pA^5 == sage_eval(repr(pF^5), A.gens_dict())
             True
         """
+        if self.data == NULL:
+            return self
         cdef PathAlgebraElement right = other
+        if right.data == NULL:
+            return right
         cdef path_homog_poly_t *H1 = self.data
         cdef path_homog_poly_t *H2
         cdef path_term_t *T2