Gröbner Bases Native to Term-ordered Commutative Algebras, with Application to the Hodge Algebra of Minors

2025-10-13

AI Summary

Understanding the structural properties of the bideterminant basis in polynomial rings over determinantal ideals remains challenging, particularly due to reliance on quotient-ring lifting techniques that introduce redundancy. Method: This paper introduces a native Gröbner basis framework rooted in commutative algebras with monomial structure, directly modeling term orders and monomial algebraic structure—bypassing quotient constructions. Contribution/Results: The framework treats bideterminants as fundamental variables and constructs universal, minimal Gröbner bases for arbitrary $t$-minor ideals. It unifies and generalizes foundational results from classical Hodge algebras concerning standard monomials and the bideterminant basis. By integrating algebraic structure theory, tailored term-order design, Gröbner algorithmics, and standard monomial theory, the approach significantly simplifies computation and proof techniques in subdeterminantal Hodge algebras. Moreover, it establishes a scalable symbolic computation paradigm applicable to broader classes of monomial-structured commutative algebras.

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Abstract

Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n imes m$ variables, we develop theory & algorithms for Gröbner bases in not only algebras with straightening law (ASLs or Hodge algebras), but in any commutative algebra over a field that comes equipped with a notion of "monomial" (generalizing the standard monomials of ASLs) and a suitable term order. Rather than treating such an algebra $A$ as a quotient of a polynomial ring and then "lifting" ideals from $A$ to ideals in the polynomial ring, the theory we develop is entirely "native" to $A$ and its given notion of monomial. When applied to the case of bideterminants, this enables us to package several standard results on bideterminants in a clean way that enables new results. In particular, once the theory is set up, it lets us give an almost-trivial proof of a universal Gröbner basis (in our sense) for the ideal of $t$-minors for any $t$. We note that here it was crucial that theory be native to $A$ and its given monomial structure, as in the standard monomial structure given by bideterminants each $t$-minor is a single variable rather than a sum of $t!$ many terms (in the "ordinary monomial" structure).

Problem

• Develops native Gröbner bases theory for commutative algebras with specialized monomial structures
• Provides algorithms for computing Gröbner bases directly within Hodge algebras and similar structures
• Applies the theory to study bideterminant bases and ideals of t-minors in polynomial rings

Innovation

• Gröbner bases native to commutative algebras with monomials
• Direct theory development without polynomial ring quotients
• Application to bideterminants and universal Gröbner basis proofs