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

k is a field of characteristic p>0, and ℓ1,…,ℓn 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 δC that encodes the “syzygy gaps” of Fq, Gq, and , for all q=pe and ai⩽q. These are close relatives of functions introduced in [P. Monsky, P. Teixeira, p-Fractals and power series—I. Some 2 variable results, J. Algebra 280 (2004) 505–536]. Like their relatives, the δC 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 δC by studying the vanishing of certain determinants.