Handle empty matrices over univariate polynomials

[Biffo89]

Fixes #39587

Adds handling of empty matrices over univariate polynomials, specifically in the following methods:

• degree() (returns -1)
• constant_matrix()
• inverse_series_trunc()
• solve_left_series_trunc()
• row_degrees() ([] if no rows, [-1, ..., -1] if no columns)
• column_degrees() (vice versa)
• is_hermite() (defaults to _is_empty_popov())
• weak_popov_form()
• right_quo_rem()
• _right_quo_rem_reduced()
• minimal_relation_basis()

📝 Checklist

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• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
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[github-actions]

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[vneiger]

Good catch. I assume that without this, this could return a matrix over QQ if self has zero rows or columns. Which could create problems if this now-rational matrix is used later in other operations.

[vneiger]

Thanks a lot for the work, this fixes the "empty matrix issue" in almost all methods. When creating #39587 I was not sure whether empty matrices should be supported by all methods (e.g. row_degrees or leading_positions), but now I do not see why not, this all seems to fit nicely.

I write "almost all" because:

sage: ring.<x> = GF(97)[]
sage: M = matrix(ring, 2, 0)
sage: B = matrix(ring, 1, 0)
sage: M.reduce(B)
---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
.....
IndexError: matrix index out of range

This should be fixed before merging.

[vneiger]

Could someone with the relevant rights allow the pending checks to run, maybe @fchapoton ? Thanks a lot.

[vneiger]

Another minor issue, to be investigated:

sage: ring.<x> = GF(97)[]
sage: mat = matrix(ring, 4, 0)
sage: mat.is_popov(include_zero_vectors=False)
True

this matrix has 4 rows and 0 columns, it should not be considered in Popov form (Popov form requires full row rank, if zero rows are not allowed). Strangely, there is no similar issue for is_hermite.

[vneiger]

Another minor issue, to be investigated:

sage: ring.<x> = GF(97)[]
sage: mat = matrix(ring, 4, 0)
sage: mat.is_popov(include_zero_vectors=False)
True

this matrix has 4 rows and 0 columns, it should not be considered in Popov form (Popov form requires full row rank, if zero rows are not allowed). Strangely, there is no similar issue for is_hermite.

In fact, this has nothing to do with the modifications in this PR. But may as well be fixed at the same time. @Biffo89 could you please replace the line

            return self._is_empty_popov(row_wise)

in both def is_popov and def is_weak_popov, by the line:

            return self._is_empty_popov(row_wise, include_zero_vectors)

? Thank you.

[vneiger]

Thanks for the fixes. In fact this also fixes the other above mentioned issue (which was a consequence of the wrong result of is_popov called in method reduce with include_zero_vectors=False):

sage: ring.<x> = GF(97)[]
sage: M = matrix(ring, 2, 0)
sage: B = matrix(ring, 1, 0)
sage: M.reduce(B)
[]
sage: M.reduce(B).dimensions()
(2, 0)

Now everything looks good to me.