Enumerate points of bounded height in projective/affine space over number fields

[sagetrac-gjorgenson]

Add functionality to iterate over points of bounded height in projective/affine space over a number field. Also add functionality to list the points of height less than a given bound on a scheme defined over a number field.

CC: @bhutz

Component: algebraic geometry

Author: Grayson Jorgenson

Branch: 63b6422

Reviewer: Ben Hutz

Issue created by migration from https://trac.sagemath.org/ticket/17386

[sagetrac-gjorgenson]

Branch: u/gjorgenson/ticket/17386

[sagetrac-git]

Commit: 456c721

[sagetrac-git]

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

456c721

Added enumeration functionality for affine space over number fields.

[bhutz]

Reviewer: Ben Hutz

[bhutz]

comment:4

Here is a first pass at reviewing this.

First you need to be careful with the difference between relative height and absolute height. The function K.elements_of_bounded_height() is returning with respect to a bound of relative height (as its documentation says). I would think most users would be interested in absolute height rational points for schemes. So you'll need to fix the functionality here, but also fix the documentation to specify which height in all appropriate places.

Also, specify whether the bound is <= or <

numberField is not a good variable name as CamelCase is reserved for class names.

In projective_homset.py put the imports near where they are needed so you only import what you need (this is done correctly in affine_homset.py)

Some examples that I expected to work that didn't

u = QQ['u'].0
K.<v> = NumberField(u^2 + 3)
A.<x,y> = ProjectiveSpace(K,1)
X=A.subscheme(x-y)
X.rational_points(3)

same for affine

from sage.schemes.projective.projective_rational_point import enum_projective_number_field
enum_projective_number_field(X,3)

from sage.schemes.affine.affine_rational_point import enum_affine_number_field
enum_affine_number_field(X,3)

I think the issues with the enum_ functions is whether the input should be the scheme X or the point set X(K). Clarify which in the documentation for the functions. However, the X.rational_points ones seem to actually be failing.

[sagetrac-git]

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

9eb56a3	17386: Implemented some of the changes discussed by the review.
9e610df	17386: Merge branch 'master' into ticket/17386
003884c	17386: finished implementing changes from first review

[sagetrac-git]

Changed commit from 456c721 to 003884c

[bhutz]

comment:7

Looks like we are almost there with this one. The functionality all is in working order. But, I came across a few minor things to fix.

affine_rational_point
example line 168 is a long test so add -> # long time
probably makes more sense to make the bound smaller and get the test < 1s

affine_space -> line 777 not needed, but move the comment to 779

affine_space -> points_of_bounded_height -> add comment that this is using the Krumm-Doyle algorithm for number fields (get the citation from number fields where they implemented it).

algebraic_scheme -> line 1502 no longer need is_RationalField(F)

projective_space -> points_of_bounded_height -> add comment that this is using the Krumm-Doyle algorithm for number fields (get the citation from number fields where they implemented it).

projective_space -> line 94 not needed -> move comment to 951

There are a few places where the lines in the docs are too long. You need to wrap those. Such as line 748 in affine_space.py

Another bad example:

u = QQ['u'].0
K.<v> = NumberField(u^2 + 3)
A.<x,y> = ProjectiveSpace(K,1)
X=A.subscheme(x-y)
X.rational_points(3)
from sage.schemes.affine.affine_rational_point import enum_affine_number_field
enum_affine_number_field(X,3)

oops...X is projective and we called enum_affine successfully!! Need a check in those functions to make sure the ambient space is Affine or Projective

[sagetrac-git]

Changed commit from 003884c to f786719

[sagetrac-git]

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

f786719

17386: Implemented changes from second review. In addition to these changes, also added checks in

[sagetrac-git]

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

63b6422

17386: removed uneeded references

[sagetrac-git]

Changed commit from f786719 to 63b6422

[bhutz]

comment:11

Ok. this looks good now.

[vbraun]

Changed branch from u/gjorgenson/ticket/17386 to 63b6422

[sagetrac-gjorgenson]

Changed commit from 63b6422 to none