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

TL;DR

The paper develops a native Gröbner-bases framework for finitely generated commutative algebras equipped with a monomial structure, termed pseudo-ASLs, by introducing algebras of leading terms $A_{lt}$ and a compatible term order. It proves key foundational results: existence and uniqueness of finite pseudo-ASL Gröbner bases, reduced and universal bases, and a Buchberger-type criterion for modules, along with Schreyer-type syzygy theory and algorithmic procedures for computing closures. The framework is then specialized to bideterminants, showing that determinants and minors naturally yield universal $bd$-Gröbner bases and that many classical results on bideterminants can be packaged within this native theory, including determinantal ideals and their universal bases. Computationally, the authors present algorithms for performing operations in pseudo-ASLs, constructing S-closures and Ann-closures, and applying the theory to Hilbert–Poincaré series and Krull dimension, illustrating the method with rank-one matrices. The approach provides a unified, intrinsic viewpoint that leverages ASL structure to achieve transparent proofs and potentially more efficient computation, with applications to determinantal varieties and Grassmannians through bideterminants.

Abstract

Motivated by better understanding the bideterminant (=product of minors) basis on the polynomial ring in $n \times m$ 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).

Theorems & Definitions (192)

• definition 1: Algebra with pseudo-straightening law, a.k.a. pseudo-ASL
• remark 1
• proof
• definition 2: Algebra with straightening law / ASL / Hodge algebra
• remark 2
• definition 3: store=deftermorder, note=Pseudo-ASL term order
• remark 3
• remark 4
• proposition 1
• proof