gh-37324: Basis completion for matrices over univariate polynomials
    
This PR (the fruit of discussions with @xcaruso during SageDays 125)
introduces a new functionality for matrices over univariate polynomials:
completing bases. A description of what this means is below: this is a
direct copy-paste of the doc strip written for the function.

Two algorithms are offered: the one which one might call "naive", based
on Smith form computation (requiring also one of the unimodular
transformations); the other one is based on an algorithm by Wei Zhou and
George Labahn ([Proceedings of ISSAC, 2014, Unimodular completion of
polynomial matrices](https://dl.acm.org/doi/10.1145/2608628.2608640)).

The default has been put to the latter, which is faster in most
circumstances, sometimes significantly (*). In fact, it does not exploit
fast polynomial multiplication (it should, but does not in the current
state of Sage), so it is the case that the Smith form variant is faster
in a restricted range of matrices of small dimensions and medium-large
degrees. If there is interest in efficiency of this computation, one
could work out some thresholds to switch between the two algorithms; but
a much more impactful move would be to incorporate fast polynomial
arithmetic in approximant bases and kernel bases, which would
dramatically speed up Zhou and Labahn's algorithm.

(*) Measured, for example: speed-up x20 for completing a 20x40 matrix of
degree 5, 10, 20, or 40; speed-up about x200 for completing a 40x80
matrix of degree 5 or 10.

Here [basis_completion.zip](https://github.com/sagemath/sage/files/14272
879/basis_completion.zip) is an archive with one file with sage code to
help if one wishes to run somewhat extensive tests as well as
benchmarks, and two files showing the results of some benchmarks.

```
def basis_completion(self, row_wise=True, algorithm="approximant"):
        Return a Smith form-preserving nonsingular completion of a basis
of
        this matrix: row-wise completing a row basis if ``row_wise`` is
True;
        column-wise completing a column basis if it is False.

        For a more detailed description, consider the row-wise case (the
        column-wise case is the same up to matrix transposition). Let
`A` be
        the input matrix, `m \times n` over univariate polynomials
        `\Bold{K}[x]`, for some field `\Bold{K}`, and let `r` be the
rank of
        `A`, which is unknown a priori. This computes a matrix `C` of
        dimensions `(n-r) \times n` such that stacking both matrices one
above
        the other, say `[[A],[C]]`, gives a matrix of maximal rank `n`
and with
        the same nontrivial Smith factors as `A`. In particular, `C` has
full
        row rank, and the rank of the input matrix may be recovered from
the
        number of rows of `C`.

        As a consequence, if `B` is a basis of the module generated by
the rows
        of `A` (for example `B = A` if `A` has full row rank), then
`[[B],[C]]`
        is nonsingular, and its determinant is the product of the
nonzero Smith
        factors of `A` up to multiplication by a nonzero element of
`\Bold{K}`.

        In particular, for `A` with full row rank: if the rows `A` can
be
        completed into a basis of `\Bold{K}[x]^{n}` (or equivalently,
`A` has
        unimodular column bases, or also, if the rows of `A` generate
all
        polynomial vectors in the rational row space of `A`), then `C`
provides
        such a completion. In this case, `[[A],[C]]` is unimodular: it
is
        invertible over `\Bold{K}[x]`, and `det([[A],[C]])` is a nonzero
        element of the base field `\Bold{K}`.
```
    
URL: #37324
Reported by: Vincent Neiger
Reviewer(s): grhkm21, Vincent Neiger, Xavier Caruso

Author: Release Manager

src/doc/en/reference/references/index.rst

@@ -6671,6 +6671,10 @@ REFERENCES:
            *The CiliPadi Family of Lightweight Authenticated Encryption*
            https://csrc.nist.gov/CSRC/media/Projects/Lightweight-Cryptography/documents/round-1/spec-doc/cilipadi-spec.pdf
 
+.. [ZL2014] Wei Zhou and George Labahn. "Unimodular completion of polynomial
+            matrices".  In Proceedings ISSAC 2014, pages 413-420. ACM, 2014.
+            :doi:`10.1145/2608628.2608640`
+
 .. [ZZ2005] Hechun Zhang and R. B. Zhang.
             *Dual canonical bases for the quantum special linear group
             and invariant subalgebras*.