Syzygy gap fractals. I: Some structural results and an upper bound (Q420714)

Let \(k\) be a field of characteristic \(p > 0\) and \(A=k[x,y]\). If \(F, G, H\) are nonzero homogeneous elements of \(A\) with no common factors, the module of syzygies of \((F,G, H)\) is free on two homogeneous generators. The syzygy gap \(\delta(F,G,H)\) of \(C=(F, G, H)\) is defined to be \(n-m\) where \(n \geq m\) are the degrees of these two generators. For a family of pairwise prime linear forms \(\ell_1,\ldots, \ell_n \in A\) and \(C=(F,G, H)\) a triple of nonzero homogeneous elements such that \(F,G\) and \(H\ell_1\cdots \ell_n\) have no common factors, the author considers the following function: let \(\mathcal{I}=[0,1] \cap \mathbb{Z}[1/p]\) and define \(\delta_C: \mathcal{I}^n \to \mathbb{Q}\) by NEWLINE\[NEWLINE\delta_C\Big(\frac{a}{q}\Big)=\frac{1}{q} \delta(F^q, G^q, H^q\ell_1^q\cdots \ell_n^q)NEWLINE\]NEWLINE where \(q=p^e\) and \(a=(a_1,\ldots, a_n) \in \mathbb{Z}^n\). As previously shown by Monsky and Teixeira, the structure of the ``syzygy gap fractals'' \(\delta_C\) is related to the explicit computation of various Hilbert-Kunz functions. In this paper the author shows that the functions \(\delta_C\) are determined by their zeros, each local maximum of \(\delta_C\) determines the behavior of the function on a certain neighborhood, and there exists an upper bound for the functions \(\delta_C\) at their local maxima.