Implement to_polynomial and from_polynomial for quasimodular forms

[DavidAyotte]

The goal of this ticket is like the ticket #32135: implement the methods to_polynomial and from_polynomial for the class QuasiModularFormElement (see #31560, #31512)

In particular, we want to be able to write a quasimodular form as a polynomial in terms of the generators of the graded algebra of quasimodular forms. Examples:

sage: QM = QuasiModularForms(1)
sage: f = QM.0 + QM.1
sage: f.to_polynomial()
E2 + E4
sage: P.<E2, E4, E6> = QQ[]
sage: g = QM.from_polynomial(E2 + E4); g
2 + 216*q + 2088*q^2 + 6624*q^3 + 17352*q^4 + 30096*q^5 + O(q^6)
sage: f == g
True

Depends on #31512

CC: @videlec

Component: modular forms

Author: David Ayotte

Branch/Commit: 6b729bc

Reviewer: Marc Mezzarobba, Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/32336

[DavidAyotte]

Branch: u/gh-DavidAyotte/quasiform_to_from_polynomial

[DavidAyotte]

Author: David Ayotte

[DavidAyotte]

Commit: 094d64a

[DavidAyotte]

Last 10 new commits:

b2517ec	updated _generators_variables_dictionnary
461d5f6	implementation of to_polynomial
1e6a262	make polynomial_ring not private, assigned degrees to the variables
7d8b20a	fix docstring of to_polynomial
e4a974f	fix docstring, removed _weights_of_gemerators, small code syntax changes
1038d68	rename find_generator.py into ring.py
d991d19	merge ticket 31559 'Make ModularFormRings manipulate formal objects' and fix conflicts
9304a66	Merge ticket 32135 'Implement to_polynomial and from_polynomial methods for ModularFormsRing' into t/31512/implementation_of_the_graded_quasimodular_forms_ring
526c884	fix constant polynomials bug
094d64a	New methods in ring.py: ngens, polynomial_ring, from_polynomial. New methods in element.py: to_polynomial, weights_list, is_homogeneous, weight, homogeneous_components.

[sagetrac-git]

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

c8fb9d1	fix some docstring. small fix.
6057a6f	fix failing doctests, pyflakes, block, tiple colon

[sagetrac-git]

Changed commit from 094d64a to 6057a6f

[sagetrac-git]

Changed commit from 6057a6f to 71ba9f0

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. Last 10 new commits:

6984a26	fix failing doctests, pyflakes, block, tiple colon
5a373a3	remove __base_ring attributes and method, changed some syntax
600fdff	fix typos
77e8a3d	add congruence subgroups support
6884d6d	add input parsing for q-expansions in _element_constructor_
31432ae	fixed input parsing for q-expansion, added documentation to reference manual, added a bibliographical reference
5ef7767	Merge branch 't/31559/make_modularformrings_manipulate_formal_objects' into t/31512/implementation_of_the_graded_quasimodular_forms_ring
377d88b	resolved merge conflicts
100f079	added missing newline, fix capitalization of title
71ba9f0	fixed merge conflicts

[sagetrac-git]

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

936b3b6

fix docbuild errors

[sagetrac-git]

Changed commit from 71ba9f0 to 936b3b6

[sagetrac-git]

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

782d722

Merge branch 'develop' into quasiform_to_from_polynomial

[sagetrac-git]

Changed commit from 936b3b6 to 782d722

[sagetrac-git]

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

d4eb6cd

fix small error

[sagetrac-git]

Changed commit from 782d722 to d4eb6cd

[mezzarobba]

comment:8

Just a minor remark: almost all other methods that convert something to a polynomial are called just polynomial, not to_polynomial. (I have no idea if there is a good reason to deviate from that here.)

[DavidAyotte]

comment:9

Thank you very much for bringing up this remark! I was not aware of this fact, as I never used polynomial conversion methods before. I agree with you: everything should be consistent. What do you think about adding an alias polynomial = to_polynomial?

I will also fix the work in #32135 to be consistent with the option we decide.

[mezzarobba]

comment:10

Replying to @DavidAyotte:

Thank you very much for bringing up this remark! I was not aware of this fact, as I never used polynomial conversion methods before. I agree with you: everything should be consistent. What do you think about adding an alias polynomial = to_polynomial?

I am not sure that the alias is necessary, but I have nothing against it!

[sagetrac-git]

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

a3245b8

fix failing doctests

[sagetrac-git]

Changed commit from d4eb6cd to a3245b8

[tscrim]

comment:12

It seems like you also have some stuff from #32357 merged in here. While I am not strictly opposed to that, I feel it would be better to have this ticket be "clean" in the sense it only has the code going to/from a polynomial ring.

I agree with Marc about calling the method polynomial. I might want to avoid the alias because it clutters things a bit. I am not that opposed to it, but I would want to call the method polynomial with to_polynomial = polynomial to subtly indicate which is the "main" method.

[DavidAyotte]

comment:13

It seems like you also have some stuff from #32357 merged in here.

Thank you very much for finding this. I think that you are talking about the method differentiation_operator (are you?). In fact, this method was written at the beginning of my GSOC project (in #31512) and it happens that I removed it in #31512, but it must have slipped through the cracks when I resolved the merge conflicts. It is indeed important to clean this!

Concerning the to_polynomial method, I will rename it as you and Marc suggested. For now, I will also leave an alias to_polynomial = polynomial.

Again, thank you very much for taking the time to look at this.

[sagetrac-git]

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

6b729bc

renamed to_polynomial into polynomial, removed differentiation_operator

[sagetrac-git]

Changed commit from a3245b8 to 6b729bc

[tscrim]

Reviewer: Marc Mezzarobba, Travis Scrimshaw

[tscrim]

comment:15

Thanks. LGTM.

[vbraun]

Changed branch from u/gh-DavidAyotte/quasiform_to_from_polynomial to 6b729bc