Values of the F-pure threshold for homogeneous polynomials

TL;DR

This paper derives a formula for the F-pure threshold of generic homogeneous polynomials and explores the range of possible thresholds for reduced polynomials, providing counterexamples, supporting evidence, and bounds across characteristics.

Contribution

It introduces a formula for the F-pure threshold of generic homogeneous polynomials and investigates the thresholds of reduced forms, including counterexamples and bounds across characteristics.

Findings

Derived a formula for the F-pure threshold in terms of n, d, and p.

Existence of reduced polynomials with thresholds as truncations of 2/d's base p expansion.

Provided a lower bound on the F-pure threshold based on degree and characteristic.

Abstract

We find a formula, in terms of n, d and p, for the value of the F-pure threshold for the generic homogeneous polynomial of degree d in n variables over an algebraically closed field of characteristic p. We also show that, in every characteristic p and for all d (greater than 3) not divisible by p, there always exist reduced polynomials of degree d whose F-pure threshold is a truncation of the base p expansion of 2/d at some place; in particular, there always exist reduced polynomials whose F-pure threshold is strictly less than 2/d. We provide an example to resolve, negatively, a question proposed by Hernandez, N\'u\~nez-Betancourt, Witt and Zhang, as to whether a list of necessary restrictions they prove on the F-pure threshold of reduced forms are "minimal" for large p. On the other hand, we also provide evidence supporting and refining their ideas, including identifying specific truncations of the base p expansion of 2/d that are always F-pure thresholds for reduced forms of degree d, and computations that show their conditions suffice (in every characteristic) for degrees up to eight and several other situations. Finally, we point out a lower bound on the F-pure threshold of a reduced form in terms of its degree and the characteristic p.