On the degrees of relations on $x_1^{d_1}, \ldots, x_n^{d_n}, (x_1+ \ldots + x_n)^{d_{n+1}}$ in positive characteristic

TL;DR

This paper derives a formula for the minimal degree of non-Koszul relations among certain monomials in positive characteristic fields, with applications to F-thresholds and Lefschetz properties of specific rings.

Contribution

It provides a new explicit formula for non-Koszul relations degrees and characterizes conditions for the weak Lefschetz property based on characteristic and exponents.

Findings

Formula for smallest degree of non-Koszul relations.

Explicit calculation of diagonal F-thresholds.

Characterization of weak Lefschetz property conditions.

Abstract

We give a formula for the smallest degree of a non-Koszul relation on $x_1^{d_1}, \ldots, x_n^{d_n}, (x_1+\ldots +x_n)^{d_{n+1}}\in k[x_1, \ldots, x_n]$ (under certain assumptions on $d_1, \ldots, d_{n+1}$) where $k$ is a field of positive characteristic $p$. As an application of our result, we give a formula for the diagonal F-threshold of a diagonal hypersurface. Another application is a characterization, depending on the characteristic $p$ of $k$, of the values of $d_1, \ldots, d_{n+1}$ (satisfying certain assumptions) such that the ring $k[x_1, \ldots, x_{n+1}]/(x_1^{d_1}, \ldots, x_{n+1}^{d_{n+1}})$ has the weak Lefschetz property.