Investigate possible inefficiency in Buchberger code

[fredrik-johansson]

The function fmpz_mpoly_buchberger_naive (and now also fmpz_mod_mpoly_buchberger_naive) is supposed to be a close to verbatim implementation of GROBNERNEW2 described in section 5.5 of Becker and Weispfenning (a citation missing in our documentation, by the way).

Back when I wrote the code, I ran some comparisons with the Groebner basis code in SymPy which is based on the same reference (https://github.com/sympy/sympy/blob/master/sympy/polys/groebnertools.py), and IIRC my code was consistently doing more steps. This is probably due to some silly mistake in the logic. @andrii302 did you figure out why this might be the case?

[rburing]

I don't know which monomial orders you tested but these lines look odd for the non-lex case:

flint/src/fmpz_mpoly/buchberger_naive.c

Line 146 in 1ad551d

if (total >= best_lcm[j])

flint/src/fmpz_mod_mpoly/buchberger_naive.c

Line 131 in 1ad551d

if (total >= best_lcm[j])

Or I guess I don't understand how that's supposed to work.

Isn't it supposed to be total >= best_lcm[nvars], as that's where the current best total is stored?

At least best_lcm[nvars] is currently unused.