F-threshold of determinantal rings

TL;DR

This paper establishes a new upper bound for the F-threshold of determinantal rings using combinatorial methods, showing it matches the a-invariant for certain matrices and conjecturing this for all cases.

Contribution

It introduces a combinatorial approach to bound the F-threshold of determinantal rings and proves equality with the a-invariant for specific matrix sizes.

Findings

F-threshold c^m(m) bounded above using combinatorics

Equality with a-invariant proven for 3×n and 4×n matrices

Conjecture that equality holds for all matrices

Abstract

In this paper, by using a combinatorial approach, we establish a new upper bound for the F-threshold $c^\mm(\mm)$ of determinantal rings generated by maximal minors. We prove that $c^\mm(\mm)$ coincides with the $a$-invariant in the case of $3\times n$ and $4\times n$ matrices and we conjecture such equality holds for all matrices.