implement critical point functionality including is_pcf for projective morphisms

[bhutz]

Implement computing the critical point locus for a morphism of projective space and checking if that morphism is post-critically finite (all critical points are preperiodic). We can also construction to critical point portrait.

Component: algebraic geometry

Author: Ben Hutz

Branch/Commit: f904d5d

Reviewer: Grayson Jorgenson

Issue created by migration from https://trac.sagemath.org/ticket/18281

[bhutz]

Branch: u/bhutz/ticket/18281

[bhutz]

Author: Ben Hutz

[bhutz]

New commits:

83242df

18281: critical point functionality for projective morphisms

[bhutz]

Commit: 83242df

[sagetrac-gjorgenson]

comment:3

change_ring (for both SchemeMorphism_polynomial and SchemeMorphism_point):

• Before creating an embedding into QQbar, need check that the map is defined over a number field, change_ring breaks otherwise:

P.<x,y> = ProjectiveSpace(ZZ,1)
H = End(P)
f = H([3*x^2,y^2])
f.change_ring(QQbar)

• There is also an issue when trying to change_ring from QQbar to QQbar:

P.<x,y> = ProjectiveSpace(QQbar,1)
H = End(P)
f = H([3*x^2,y^2])
f.change_ring(QQbar)

_preperiodic_points_to_cyclegraph:

• maybe add more comments explaining how equal points with different representations are dealt with

wronskian_ideal:

• in line 2497, change format of NotImplementedError to the standard format
• change "== false" checks to "not" for faster performance

critical_points:

• rational_points should return a list, so the list conversion in the return statement is not needed

is_postcritically_finite

• remove extra lines in documentation, lines 2560 and 2595
• subscheme check not needed since the check is present in critical_points
• perhaps restrict to only maps defined over number fields

critical_point_portrait:

• copying crit_points list not needed, instead could loop over crit_points by index, stopping when the index is the original length of the crit_points list
• subscheme and endomorphism checks not needed since they are present in critical_points

critical_height:

• maybe include a note that the keywords are optional
• subscheme and endomorphism checks are already done in critical_points

[sagetrac-gjorgenson]

Reviewer: Grayson Jorgenson

[sagetrac-git]

Changed commit from 83242df to 5a9f935

[sagetrac-git]

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

5a9f935

18281: changes from review

[bhutz]

comment:5

I made the changes. I changed the descriptions of the kwds for heights in a number of places.

I also decided that critical heights over number field orders was ok and moved the function to the 'rings' section.

[sagetrac-gjorgenson]

comment:6

I tested is_postcritically_finite and critical_point_portrait on examples of pcf maps.

Everything looks good, but there is one instance of whitespace in line 2184 of projective_morphism

[sagetrac-git]

Changed commit from 5a9f935 to f904d5d

[sagetrac-git]

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

f904d5d

18281 removing trailing white space

[vbraun]

Changed branch from u/bhutz/ticket/18281 to f904d5d