Add is_polynomial and normal_form to projective morphism

[sagetrac-rlmiller]

Added function to determine whether an equation is a polynomial or not by checking if it has a totally ramified fixed point. (is_polynomial)

Then used this function to create make_look_poly, which puts polynomials in the form "xn + a*x(n-2) +...+c", where a and c are constants.

Needs more examples, specifically for errors.

CC: @bhutz

Component: algebraic geometry

Author: Rebecca Lauren Miller, Ben Hutz

Branch/Commit: 2da7679

Reviewer: Ben Hutz, Rebecca Lauren Miller

Issue created by migration from https://trac.sagemath.org/ticket/20650

[sagetrac-rlmiller]

Branch: u/rlmiller/polynomials

[sagetrac-rlmiller]

Commit: 753f9ad

[sagetrac-rlmiller]

New commits:

753f9ad

20650 Created is_polynomial and make_look_poly functionc

[bhutz]

comment:5

• The documentation does not build

[schemes  ] /home/ben/sage/sage-test/local/lib/python2.7/site-packages/sage/schemes/projective/projective_morphism.py:docstring of sage.schemes.projective.projective_morphism.SchemeMorphism_polynomial_projective_space_finite_field.make_look_poly:13: ERROR: Unexpected indentation.
Error building the documentation.

• Totally Rmified Fixed Point does not need initial caps.

• "xn + a*x(n-2) +...+c" should not be "

• the examples sage:: should only have one :.

• the doc tests pass, but only because the examples with :: are not being run.

• formatting of the code needs some attention, such as spaces around =

• these two functions were added to the finite field section of polynomial morphism. Should they work over other rings/fields as well?

• make_look_poly could use a better name. How about 'normal_form'? Since it has only one keyword, you should use the keyword instead of the dict kwds

[sagetrac-git]

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

c19032f

20650 fixed documentaion

[sagetrac-git]

Changed commit from 753f9ad to c19032f

[bhutz]

Changed branch from u/rlmiller/polynomials to u/bhutz/polynomials

[bhutz]

comment:9

I did some code clean-up. No functionality was changed except that instead of having to compute the inverses of the conjugation it now just computes the right one to start with.

There are still a few functionality issues: Other base rings that should work do not such as QQ

P.<x,y>=ProjectiveSpace(QQ,1)
H=End(P)
f=H([x^2+y^2,y^2])
f.is_polynomial()

I think it likely these will work for finite fields as well

P.<x,y>=ProjectiveSpace(GF(13),1)
H=End(P)
f=H([x^2+y^2,y^2])
f.is_polynomial()

Function field base rings are not going to work, but this error isn't very informative

R.<c>=PolynomialRing(QQ)
P.<x,y>=ProjectiveSpace(FractionField(R),1)
H=End(P)
f=H([x^2+c*y^2,y^2])
f.is_polynomial()

R.<c>=FunctionField(QQ)
P.<x,y>=ProjectiveSpace(R,1)
H=End(P)
f=H([x^2+c*y^2,y^2])
f.is_polynomial()

The single rational preimage check is not sufficient

K.<w>=QuadraticField(4/27)
P.<x,y>=ProjectiveSpace(K,1)
H=End(P)
S=P.coordinate_ring()
f=H([x^3+w*y^3,x*y^2])
f.is_polynomial()

---

New commits:

530a585

20650: code clean up

[bhutz]

Changed commit from c19032f to 530a585

[bhutz]

Reviewer: Ben Hutz

[bhutz]

comment:10

One more thing: in dimension > 1 it should return a NotImplmentedError

[sagetrac-rlmiller]

Changed branch from u/bhutz/polynomials to u/rlmiller/polynomials

[sagetrac-rlmiller]

Changed commit from 530a585 to f913bde

[sagetrac-rlmiller]

New commits:

f913bde

20650 Fixed errors with finite field and QQ

[bhutz]

Changed branch from u/rlmiller/polynomials to u/bhutz/polynomials

[bhutz]

comment:14

Going all the way to the splitting field was causing some slow performance, so I've reworked it to manage the fields directly. This includes special casing finite fields and managing the embeddings as well.

it should be noted that a simple error in projective_morphism.conjugation() was corrected in this ticket as well as a simple error in affine_morphism.homogenize()

The new version is ready for review now.

---

New commits:

2da7679

20650: rework functions for speed, finite fields

[bhutz]

Changed author from Rebecca Lauren Miller to Rebecca Lauren Miller, Ben Hutz

[bhutz]

Changed commit from f913bde to 2da7679

[sagetrac-rlmiller]

Changed reviewer from Ben Hutz to Ben Hutz, Rebecca Lauren Miller

[vbraun]

Changed branch from u/bhutz/polynomials to 2da7679