Implement power_sums and power_sums_to_poly

[videlec]

After flint pull request 213 and flint pull request 221 flint will have a native support for going back and forth between an integer polynomial and its power sums series.

We provide generic methods power_sums and power_sums_to_poly for any polynomial and use direct call to flint when available.

Component: number fields

Issue created by migration from https://trac.sagemath.org/ticket/20004

[videlec]

Description changed:

--- 
+++ 
@@ -1,3 +1,3 @@
-After [flint pull request 213](https://github.com/wbhart/flint2/pull/213) flint will have a native support for going back and forth between an integer polynomial and its series given by power sums.
+After [flint pull request 213](https://github.com/wbhart/flint2/pull/213) and [flint pull request 221](https://github.com/wbhart/flint2/pull/221) flint will have a native support for going back and forth between an integer polynomial and its power sums series.
 
 We provide generic methods `power_sums` and `power_sums_to_poly` for any polynomial and use direct call to flint when available.

[kedlaya]

comment:2

When FLINT is not applicable, power_sums can also try calling PARI's polsym, although I don't think the implementation therein is anything special.

[videlec]

comment:3

Replying to @kedlaya:

When FLINT is not applicable, power_sums can also try calling PARI's polsym, although I don't think the implementation therein is anything special.

Indeed, it uses Newton formula. However, there is #20399 to review first because

sage: x = polygen(ZZ)
sage: p = x^2 + 4
sage: c = p._pari_().polsym(5)
sage: c
[2, 0, -8, 0, 32, 0]~
sage: c.sage()
Traceback (most recent call last):
...
TypeError: Object (=[2, 0, -8, 0, 32, 0]~) must be of type t_VEC.

[videlec]

Changed dependencies from #18356 to #20399

[kedlaya]

comment:4

Now that #20399 is resolved, maybe time for another look?

[videlec]

comment:5

1. Indeed, the pari way does now work

sage: x = polygen(ZZ)
sage: p = x^2 + 4
sage: c = p.__pari__().polsym(5)
sage: c
[2, 0, -8, 0, 32, 0]~
sage: c.sage()
[2, 0, -8, 0, 32, 0]

2. Concerning the original goal of the ticket, there is no release of flint yet...

[videlec]

Changed dependencies from #20399 to none

[kedlaya]

comment:6

Did this make it into FLINT 2.6.0? If so, we should add #29719 as a dependency.