\(p\)-fractals and power series. I: Some 2 variable results
Let \(A= kx_1,\dots, x_s\) be a power series ring in \(s\) variables over a field \(k\) of characteristic \(p> 0\). Let \(I\) be an ideal of \(A\). The degree of \(I\), denoted by \(\deg I\), is the \(k\)-dimension of \(A/I\). Let \(I^{[q]}\) denote the ideal generated by all \(z^q\), \(z\in I\). Then \(\deg I^{[q]}= q^s\deg I\). The authors consider the following two questions which are related to the theory of the Hilbert-Kunz function: (1) Suppose \(\deg I<\infty\). How does \(\deg(I^{[q]},\prod^r_{i=1} h^{a_i}_i)\), \(0\leq a_i\leq q\), depend on \(a_1,\dots, a_r\)? (2) Suppose \(\deg(u_1,\dots, u_r)< \infty\). How does \(\deg(u^{a_1}_1,\dots, u^{a_r}_r)\) depend on \(a_1,\dots, a_r\)? Let \({\mathcal I}\) denote the set of all rational numbers in \([0,1]\) whose denominators are powers of \(p\) and let \(\mathbb{Q}\) denote the set of all rational numbers. Define the functions \(\varphi_I,\varphi_u:{\mathcal I}^r\to \mathbb{Q}\) as follows: (1) \(\varphi_I(a_1/q,\dots, a_r/q)= q^{-s}\deg(I^{[q]},\prod^r_{i=1} h^{a_i}_i)\); (2) \(\varphi_u(a_1/q,\dots, a_r/q)= q^{-s}\deg(u^{a_1}_1,\dots, u^{a_r}_r)\). If \(r= s\), then \(\deg(u_1^{a_1},\dots, u^{a_r}_r)= \deg(u_1,\dots, u_r)\prod^r_{i=1} a_i\). If \(s= 1\), then both questions above can be answered easily. Let \(r\) be a fixed positive integer and let \({\mathcal F}^r\) be the \(\mathbb{Q}\)-algebra of functions \(\varphi\:{\mathcal I}^r\to\mathbb{Q}\). Let \(\varphi\in{\mathcal F}^r\) and let \(b= (b_1,\dots, b_r)\) be an integer vector with \(0\leq b_i< q\). Let \(T_{q|b}\varphi\) denote \((t_1,\dots, t_r)\mapsto\varphi((t_1+ b_1)/q,\dots, (t_r+ b_r)/q)\). Then \(T_{q|b}\varphi\in{\mathcal F}^r\). \(\varphi\) is called a \(p\)-fractal (of dimension \(r\)) if all the \(T_{q|b}\varphi\), \(q\) a power of \(p\), \(0\leq b_i< q\), span a finite-dimensional \(\mathbb{Q}\)-subspace \(V\) of \({\mathcal F}^r\). The authors give answers for the two questions above for the simplest nontrivial cases in the following results: Theorem 1. Suppose \(A= kx, y\) with \(k\) finite, and \(\deg I<\infty\). Then the function \(\varphi_I:{\mathcal I}^r\to \mathbb{Q}\), \((a_1/q,\dots, a_r/q)\mapsto q^{-2}\deg(I^{[q]}, \prod^r_{i=1} h^{a_i}_i)\), is a \(p\)-fractal. Theorem 2. Suppose \(A\) is as above and \(\deg(u_1,u_2,u_3)< \infty\). Then the function \(\varphi_u:{\mathcal I}^r\to \mathbb{Q}\), \((a_1/q, a_2/q, a_3/q)\mapsto q^{-2}(u^{a_1}_1, u^{a_2}_2, u^{a_3}_3)\), is a \(p\)-fractal.
scientific article; zbMATH DE number 2136734