LaurentPolynomialRing define the zero polynomial to have degree -1 which is ambiguous

[GiacomoPope]

Steps To Reproduce

sage: R = LaurentPolynomialRing(QQ, 3, 'x')
sage: x0, x1, x2 = R.gens()
sage: R.zero().degree()
-1
sage: R.zero().degree(x0)
-1
sage: R.zero().degree(x1)
-1
sage: R.zero().degree(x2)
-1
sage: (1/x0).degree(x0)
-1
sage: (1/x0).degree()
-1

Expected Behavior

The degree of 0 should be distinct from polynomials we can construct

Actual Behavior

The degree of 0 is -1, the same as 1/x0 or other similar polynomials.

Additional Information

For standard univariate and multivariate polynomial rings, the degree being -1 is essentially a nice trick in place of $-\infty$, but for Laurent polynomial rings where negative degree makes total sense, this ambiguity with zero is confusing.

One idea would be to instead have it return - infinity.

Environment

- **Sage Version**: SageMath version 10.3.rc0, Release Date: 2024-02-25

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide

[GiacomoPope]

Some further context, we also have:

sage: R.<x> = LaurentSeriesRing(QQ)
sage: R.zero().valuation()
+Infinity
sage: R.zero().degree()
-1
sage:
sage: R.<x> = LaurentPolynomialRing(QQ)
sage: R.zero().valuation()
+Infinity
sage: R.zero().degree()
-1
sage:
sage: R.<x> = LazyLaurentSeriesRing(QQ)
sage: R.zero().valuation()
+Infinity
sage: R.zero().degree()
# Errors

So the valuation seems to have been written with this in mind but the degree is picking up the result from the underlying polynomial ring? Makes me lean towards a patch giving -Infinity for the zero polynomial

Discussion: https://groups.google.com/g/sage-devel/c/XVz-4kqWuuo