F-threshold functions: Syzygy gap fractals and the two-variable homogeneous case. (May 2017) - Hernández, Daniel J.; Teixeira, Pedro
Abstract: In this article we study F -pure thresholds (and, more generally, F -thresholds) of homogeneous polynomials in two variables over a field of characteristic p > 0 . Passing to a field extension, we factor such a polynomial into a product of powers of pairwise prime linear forms, and to this collection of linear forms we associate a special type of function called a syzygy gap fractal . We use this syzygy gap fractal to study, at once, the collection of all F -pure thresholds of all polynomials constructed with the same fixed linear forms. This allows us to describe the structure of the denominator of such an F -pure threshold, showing in particular that whenever the F -pure threshold differs from its expected value its denominator is a multiple of p . This answers a question of Schwede in the two-variable homogeneous case. In addition, our methods give an algorithm to compute F -pure thresholds of homogenous polynomials in two variables.