Syzygy gap fractals--I. Some structural results and an upper bound

Abstract

k is a field of characteristic p>0, and l_1,...,l_n are linear forms in k[x,y]. Intending applications to Hilbert--Kunz theory, to each triple C=(F,G,H) of nonzero homogeneous elements of k[x,y] we associate a function delta_C that encodes the "syzygy gaps" of F^q, G^q, and H^q*l_1^{a_1}*...*l_n^{a_n}, for all q=p^e and a_i<= q. These are close relatives of functions introduced in "p-Fractals and power series--I" [P. Monsky, P. Teixeira, p-Fractals and power series--I. Some 2 variable results, J. Algebra 280 (2004) 505--536]. Like their relatives, the delta_C exhibit surprising self-similarity related to "magnification by p," and knowledge of their structure allows the explicit computation of various Hilbert--Kunz functions. We show that these "syzygy gap fractals" are determined by their zeros and have a simple behavior near their local maxima, and derive an upper bound for their local maxima which has long been conjectured by Monsky. Our results will allow us, in a sequel to this paper, to determine the structure of the delta_C by studying the vanishing of certain determinants.