Enhanced Algorithms For F-Pure Threshold Computation

Abstract

We explore different computational techniques for the F-pure threshold invariant of monomial ideals and of polynomials. For the former, we introduce a novel algorithm to reduce the number of generators of the ideal and the number of variables involved in the remaining generators, thus effectively creating a new ``simpler'' ideal with the same value of the F-pure threshold. Then, the value is the sum of entries of the inverse to the new ideal's splitting matrix. This algorithm can be further improved by using the integral closure of the ideal.

For polynomials, we introduce a direct computational technique involving properties of roots of Deuring polynomials, which are closely related to Legendre polynomials. This technique is then applied to two different families of polynomials: polynomials defining Elliptic Curves, and bivariate homogeneous polynomials with up to four distinct roots in projective space of dimension 1. The invariance of the F-pure threshold under changing variables is then used to prove properties of prime characteristic roots of Legendre polynomials.

We end the dissertations with generalizing the Deuring polynomial techniques used thus far, and introducing a way to explicitly stratify the coefficient space of polynomials supported by a fixed set of monomials, by identifying regions representing polynomials with the same F-pure threshold. We give an explicit description of the different strata as subschemes of a projective space.