Height of a dynamical system is wrong

[guojing0]

The height of a dynamical system is currently calculated as the max of the heights of the coefficients, which is incorrect. To avoid writing a complicated formula, the correct height can be thought of as treating the coefficients of all coordinate functions as one big projective point and taking the height of that. For example

P.<x,y>=ProjectiveSpace(QQ,1)
f=DynamicalSystem([1/25*x^2 + 25/3*x*y+y^2,1*y^2])
exp(f.global_height())

gives 25, but should be 625.

CC: @bhutz @EnderWannabe

Component: dynamics

Keywords: gsoc2022

Author: Jing Guo

Branch/Commit: d24eea8

Reviewer: Ben Hutz

Issue created by migration from https://trac.sagemath.org/ticket/33971

[guojing0]

Branch: u/gh-guojing0/33971_dyn_sys_height

[guojing0]

Commit: 5fb2a6e

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

a2892e0

polynomial_element_generic.py: Add `height` fn

[sagetrac-git]

Changed commit from 5fb2a6e to a2892e0

[sagetrac-git]

Changed commit from a2892e0 to 0d58271

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0d58271

multi_polynomial_element.py and polynomial_element_generic.py: Implement

[EnderWannabe]

comment:7

These changes are meant to add a height function for polynomials. As such, they should have their own ticket, as this ticket is meant to address dynamical systems specifically.

[guojing0]

comment:8

As suggested by Alex, a new ticket #34060 is created.

[guojing0]

Changed commit from 0d58271 to none

[guojing0]

Changed branch from u/gh-guojing0/33971_dyn_sys_height to none

[guojing0]

Branch: u/gh-guojing0/33971_ds_height

[guojing0]

Commit: 38425b5

[guojing0]

New commits:

38425b5

projective_morphism.py: Debug `global_height` fn

[sagetrac-git]

Changed commit from 38425b5 to c7187f4

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

c7187f4

projective_morphism.py: New example and move `import` inside

[guojing0]

comment:11

The return values of the examples in the docstring of global_height in projective_morphism.py are same as before, except for the first one (current result is something like 20.83484...):

sage: P.<x,y> = ProjectiveSpace(QQ,1)
sage: H = Hom(P,P)
sage: f = H([1/1331*x^2+1/4000*y^2, 210*x*y]);
sage: f.global_height()
8.29404964010203

I also added the example from description to the documentation.

[bhutz]

comment:13

A couple issues here

• The dimension of the projective space is off by 1. It should be

P = ProjectiveSpace(K, len(coeffs)-1)

• I'd suggest having one less variable defined

proj_point = P.point(coeffs)
return proj_point.global_height()

to

return P.point(coeffs).global_height()

• correct the doc string to describe what this height is actually doing. (i.e., this is the global height of the coefficents as a projective point)

• There seems to be an issue with scaling. The definition of height should be independent of the representative, so these should return the same height. I suggest adding an example for this too. There is an issue somewhere as these are giving different values.

P.<x,y>=ProjectiveSpace(QQ,1)
f=DynamicalSystem([1/25*x^2 + 25/3*x*y+y^2,1*y^2])
print(f.global_height())
c=10000
f.scale_by(c)
f.global_height()

• These should be implemented for Affine Dynamical systems too. Careful here that you construct the correct projective point; it may be simplest to homogenize the dynamical system first and then call the projective version (don't forget the case of morphisms between spaces of different dimensions).

• The local_height and local_height_arch seem not to be implemented for affine morphisms at all yet.

[bhutz]

Reviewer: Ben Hutz

[bhutz]

comment:20

• add description to the affine one that you are taking the height of the homogenization.

-doc test failures:

1 item had failures:
   3 of  32 in sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space.global_height
    [612 tests, 3 failures, 4.79 s]

I did not run the full set of doc tests, so be sure to do so.

• docs fail to build

OSError: /home/ben/sage-dev/local/var/lib/sage/venv-python3.10.3/lib/python3.10/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space.global_height:60: WARNING: Literal block expected; none found.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0a61770

Correct doc

[sagetrac-git]

Changed commit from 28369c6 to 0a61770

[sagetrac-git]

Changed commit from 0a61770 to 94a2805

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

94a2805

33971: Correct doc

[guojing0]

comment:23

The status badge says that building documentation is successful. It passed all the tests after I ran ./sage -t affine_morphism.py and ./sage -t projective_morphism.py.

[sagetrac-git]

Changed commit from 94a2805 to ab35cbc

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

ab35cbc

33971: Correct `prec` and relevent examples

[bhutz]

comment:25

The functionality works when the domain and codomain have the same dimension, but not when they do not.

P.<x,y,z> = ProjectiveSpace(QQ,2)
A.<z,w>=ProjectiveSpace(QQ,1)
H = Hom(P,A)
f = H([1/1331*x^2 + 4000*y*z,y^2])
f.global_height()

P.<x,y> = AffineSpace(QQ,2)
A.<z>=AffineSpace(QQ,1)
H = Hom(P,A)
f = H([1/1331*x^2 + 4000*y])
f.global_height()

This example no longer makes sense since normalization does not have an effect on the height. Delete the text and the second half of the example.

        This function does not automatically normalize::

            sage: P.<x,y,z> = ProjectiveSpace(ZZ,2)
            sage: H = Hom(P,P)
            sage: f = H([4*x^2 + 100*y^2, 210*x*y, 10000*z^2]);
            sage: f.global_height()
            8.51719319141624
            sage: f.normalize_coordinates()
            sage: f.global_height()
            8.51719319141624

[sagetrac-git]

Changed commit from ab35cbc to d24eea8

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

d24eea8

33971: Allow domain and codomain have diff dim's

[guojing0]

comment:27

The two examples are added as parts of the EXAMPLES.

[bhutz]

comment:28

This is passing my tests now. I'd like to have the patchbot pick it up before I marked it positive. Let's give it a couple days.

[vbraun]

Changed branch from u/gh-guojing0/33971_ds_height to d24eea8