Improvements to squarefree_decomposition() for finite fields.

[tscrim]

📚 Description

We make some optimizations to the computation of squarefree_decomposition() for polynomials over finite fields. This is a followup to #35323.

We make improvements by only getting the parent of T0 once and manually keep track of its degree.

The current branch

sage: x = GF(4)["x"].gen()
sage: p = x^2 + 1
sage: %timeit p.squarefree_decomposition()
140 µs ± 22.1 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

versus with #35323:

168 µs ± 5.28 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

versus 10.0.beta5:

158 µs ± 6.74 µs per loop (mean ± std. dev. of 7 runs, 10,000 loops each)

📝 Checklist

• I have made sure that the title is self-explanatory and the description concisely explains the PR.
• I have linked an issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation accordingly.

⌛ Dependencies

• Fix squarefree_decomposition failure over GF(2) #35323 Based on this change and needs to avoid conflicts.

[tscrim]

It's amazing what a little bit of Cython and avoiding unnecessary calls can do. :)

[remyoudompheng]

Hello, the code change looks OK. By the way I am finalizing a patch to link with NTL for factor and squarefree_decomposition so that only sparse polynomials (anything else?) will rely on the Python implementation.

[tscrim]

Feel free to put me as a reviewer once that PR is ready.

[github-actions]

Documentation preview for this PR is ready! 🎉
Built with commit: fdc51c2

[tscrim]

All tests pass.

@remyoudompheng Did you intend to approve this PR or should this wait for your upcoming change?

[remyoudompheng]

LGTM. The upcoming change is independent.

[tscrim]

Thank you. Can you also approve the PR? In principle I can do it, but it feels a bit weird.

[tscrim]

Thanks!