F-threshold functions: Syzygy gap fractals and the two-variable homogeneous case

详细信息    查看全文

• 作者：Daniel J. Herná ; ndez^a ; ^{dhernan@math.utah.edu} ; Pedro Teixeira^b ; ^{pteixeir@knox.edu}
• 关键词：F-pure threshold ; F-threshold ; Syzygy gap
• 刊名：Journal of Symbolic Computation
• 出版年：2017
• 出版时间：May-June 2017
• 年：2017
• 卷：80
• 期：part_P2
• 页码：451-483
• 全文大小：2079 K
• 卷排序：80

文摘

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>0p>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.