Meta-ticket: Use new features from FLINT 2.x, 3.x

[mkoeppe]

As of Sage 10.4, Sage ships FLINT 3.0.1 (update to 3.1.3 in #37203), accepts system FLINT >= 3.0.

Many of the features added in the FLINT 2.x and 3.x series are so far unused in Sage.

Tickets/features:

• matrix classes for flint polynomials #8939
• Wrap flint's fast rational functions over QQ #16429
• Add a finite field implementation using FLINT's fq and fq_nmod modules #16664
• Implement power_sums and power_sums_to_poly #20004
• add minimal interface for using ZZ[x]-matrices from flint #29894
• Wrap FLINT 2.6 functions, drop support for system FLINT < 2.6 #31525
• Support FLINT 2.8.0 #32175
• Create matrix class for FLINT's nmod_mat module #31548 Use FLINT's nmod_mat_t for matrices mod n.
• Multivariate polynomials with FLINT #31666 very fast multi-variate polynomials - use them for Sage polynomial rings and perhaps in sage.tensor.modules
• Flint generic rings #36433

For creating the bindings to the new FLINT functions, there is a potential for synergy with https://github.com/fredrik-johansson/python-flint; the SymPy developers have announced (2021) that they plan to use python-flint.

CC: @slel @videlec @fredrik-johansson @isuruf

Component: algebra

Keywords: flint

Issue created by migration from https://trac.sagemath.org/ticket/31408

[slel]

Changed keywords from none to flint

[slel]

comment:4

Possibly related: #16812, #16664, #16429.

Older: #9541, #8939.

[mkoeppe]

comment:6

Sage development has entered the release candidate phase for 9.3. Setting a new milestone for this ticket based on a cursory review of ticket status, priority, and last modification date.