Fixing bugs in the exterior algebra Gröbner basis F4 algorithm implementation

[tscrim]

This fixes serious bugs in the computation of the Gröbner basis of exterior algebra:

• not doing all proper reductions (thanks to Michael Cuntz to pointing out);
• wrong type of Gröbner basis (as defined by Stokes), which reduces uniquely but does not work for ideal containment;
• modifying internal states of elements during specific reduction steps (i.e., they are not acting like immutable (hence, hashable) elements).

We rework the implementation to address these (and hopefully some optimizations).

📝 Checklist

• The title is concise and informative.
• 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 and checked the documentation preview.

⌛ Dependencies

[tscrim]

@fchapoton Can you review this? I know this is a bit outside your usual expertise, but I'm happy to help explain stuff. This is a fairly serious bug silently returning wrong answers, so I want to get it reviewed and merged quickly.

[github-actions]

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[fchapoton]

I can try, but maybe not in depth. You introduced a TAB in the reference file, near Stokes, I think

[tscrim]

Thank you. Fixed.

[fchapoton]

Some stupid questions:

• what is the leading monomial ? in top degree ?
• what is the semantic of .ls and .lsi ?
• you are using both subsets as bitsets and some kind of canonical numbering by integers, right ?

I am afraid that I will not be able to check the math, sorry. Given that the tests pass, I am ready to give a positive review based on trust and my superficial reading. Would it be enough for you ?

[tscrim]

The leading monomial/support/coefficient is the usual term from Groebner theory, where a particular term ordering is taken (here only degrevlex, deglex, neglex are implemented). The leading support is stored in ls, with the corresponding integer being lsi in the (finite) ordering of monomials. (Both are stored for faster comparisons, look-up, and indexing.)

Honestly, I don't expect someone to fully go through the math as I had to do some for this as it doesn't appear to be in the literature in this form AFAIK. The implementation mostly uses the ideas from the classical F4 algorithm taken with making sure it expands the GB by things that remove the leading term (which doesn't happen in the classical theory). If this and the passing doctests are sufficient for you, then let's get this in as this now covers cases that were broken before.

[fchapoton]

ok, looks good enough and the tests pass locally

[tscrim]

Thank you very much.

[dimpase]

I wonder in which dimensions $d$ (i.e. number of variables) one has to work to get Groebner basis approach more efficient than the brute force linear algebra (as, unlike in the polynomial rings, everything is finite-dimensional, i.e. a quotient of the dimension $2^d$ module).

[tscrim]

That's a good question. By brute-force-linear-algebra, you mean construct a basis in echelon form for the whole ideal, correct? My guess is something like $d \leq 5$. Definitely need to optimize both implementations first...

[dimpase]

That's a good question. By brute-force-linear-algebra, you mean construct a basis in echelon form for the whole ideal, correct? My guess is something like d ≤ 5 . Definitely need to optimize both implementations first...

This depends on the field you are a lot.
Over a finite field you can do $d\le 15$ - and this probably means that over the rationals you can go close to this with a multimodal algorithm