Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms

TL;DR

This paper computes the $F$-pure threshold for certain bivariate degree four polynomials using Legendre polynomial roots and explores their connection to an open question relating $F$-pure and log canonical thresholds.

Contribution

It provides a direct method to compute the $F$-pure threshold for specific bivariate forms and links these computations to properties of Legendre polynomials over finite fields.

Findings

Explicit formula for $F$-pure threshold of degree four forms

Connection between roots of Legendre polynomials and $F$-pure thresholds over $ield_p$

Insight into the relationship between $F$-pure and log canonical thresholds

Abstract

We provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomial in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific M\"{o}bius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the $F$-pure and the log canonical threshold, and roots of Legendre polynomials over $\mathbb{F}_p$.