Groebner bases via linkage

TL;DR

This paper establishes a new criterion for identifying Gr"obner bases based on linkage patterns, and applies it to ideals generated by minors and pfaffians, proving their Gr"obner basis properties and algebraic features.

Contribution

It introduces a linkage-based criterion for Gr"obner bases and demonstrates its effectiveness on ideals generated by minors and pfaffians in matrices and ladders.

Findings

Minors and pfaffians form reduced Gr"obner bases under certain conditions.

Initial ideals are Cohen-Macaulay, squarefree, and vertex decomposable.

The approach is algebraic and combines liaison theory with Hilbert function computations.

Abstract

In this paper, we give a sufficient condition for a set $\mathal G$ of polynomials to be a Gr\"obner basis with respect to a given term-order for the ideal $I$ that it generates. Our criterion depends on the linkage pattern of the ideal $I$ and of the ideal generated by the initial terms of the elements of $\mathcal G$. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix or the ladder. We use the sufficient condition that we established to prove that the minors or pfaffians form a reduced Gr\"obner basis for the ideal that they generate, with respect to any diagonal or anti-diagonal term-order. We also show that the corresponding initial ideal is Cohen-Macaulay and squarefree, and that the simplicial complex associated to it is vertex decomposable, hence shellable. Our proof relies on known results in liaison theory, combined with a simple Hilbert function computation. In particular, our arguments are completely algebraic.