F-threshold functions: Syzygy gap fractals and the two-variable homogeneous case. (May 2017)

• Record Type: Journal Article
• Authors: Hernández, Daniel J.; Teixeira, Pedro
• Is Part Of: Journal of symbolic computation. Volume 80:Part 2(2017)
• Page Start: 451
• Page End: 483
• Publication Date: 2017-05
• Subjects: F-pure threshold -- F-threshold -- Syzygy gap; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2016.07.003
• Languages: English
• ISSNs: 0747-7171

Abstract

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.