Compute the characteristic varieties of a finitely presented group

[enriqueartal]

Let $G$ be a finitely presented group. For a field $\mathbb{k}$, let $\Lambda:=\mathbb{k}[G]$ be its group algebra and let $\mathbb{T}$ be its maximal spec, which is a torus. An element $\xi\in\mathbb{T}$ determines a local system of coefficients and the characteristic varieties of $G$ are defined as
$$V_k(G):=\{\xi\in\mathbb{T}\mid \dim H^1(G;\xi)\geq k\}.$$
Except for the neutral element these are the reduced varieties of the Fitting ideals of the Alexander matrix of the presentation. The function char_var gives systems of generators of these ideals. A groebner option is given to give Groebner basis; it is set as an option since the computations may be quite long.

📝 Checklist

• The title is concise, informative, and self-explanatory.
• 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.
• I have updated the documentation accordingly.

⌛ Dependencies

[tscrim]

Suggested change

	res0 = [[S(_) for _ in J.gens()] for J in res]
	res0 = [[S(g) for g in J.gens()] for J in res]

[enriqueartal]

I changed it also.

[tscrim]

Mirco-optimization: pull out the [(S.one(),)] into a variable to avoid reconstructing the list and tuple every loop.

[tscrim]

@enriqueartal Something is horribly broken with the bot. Please don't try anything further. I have already reported it.

[enriqueartal]

I realized that monomial_reduction gave a polynomial and an integer in the univariate case and a polynomial and a monomial in the multivariate case. Now, it is always the second case. Sorry for the mistake.

[tscrim]

Good catch! How did you come across this? Is it simple to add a doctest about it somewhere? (By this, I mean not in the method itself, but instead in, say, called via the char variety method.)

I have one small way to improve the efficiency of the method too.

[tscrim]

Suggested change

	u, n = self.polynomial_construction()
	return (u, self.parent().gen(0) ** n)
	return (self.__u, self._parent.gen(0) ** self.__n)

Since you're in the class itself.

[enriqueartal]

Thanks for the tip, changed. Actually there was a doctest for the univariate case; since the code only uses the polynomial output of monomial_reduction the difference between the output of the methods in the uni- and multi-variate case didn't affect. Anyway, I found it strange.
I came into it because I was thinking about some improvements for Laurent polynomials (outside this PR), in order to apply later PR #36299. Actually there are quite a lot of things to do; for example, univariate Laurent polynomial rings have not the polynomial_ring method (as multivariate have), or they do not know to be PID when the base ring is a field.

[tscrim]

Thanks.

Yes, there are lots of ways Laurent polynomials can be improved (and likely other methods that are not in sync between the uni- and multi-variate implementations).

Then back to positive review. Feel free to cc me on any subsequent tickets, I will try to review them when I have a chance.

[github-actions]

Documentation preview for this PR (built with commit 7a7c564; changes) is ready! 🎉