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Buchberger-like criterion without non-zerodivisors

Establish a Buchberger-like criterion for pseudo-ASL Gröbner bases (as in Theorem 4.* of the paper) that does not require assuming the leading terms of the basis elements are non-zerodivisors.

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Background

The present pseudo-ASL analogue of Buchberger’s criterion assumes that the leading terms are non-zerodivisors (e.g., holds when A is a domain). Removing this hypothesis would broaden applicability, but technical obstacles remain in the current framework.

References

It is plausible to us that the condition in Theorem~\ref{thm:buchberger2} that the leading terms of $G$ be non-zerodivisors could be dropped with some additional work. Although we are not as yet able to prove this, here we develop some preparatory lemmata that are headed in this direction, and end this section with a remark sketching some ideas and obstacles towards such a more general result.

— Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors (2510.11212 - Grochow et al., 13 Oct 2025) in Section 4.3 Towards dropping the nonzerodivisor condition