p-Fractals and power series—II.: Some applications to Hilbert–Kunz theory

We use the results of our paper 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] to prove the rationality of the Hilbert–Kunz series of a large family of power series, including those of the form ∑ifi(xi,yi), where the fi(xi,yi) are power series with coefficients in a finite field. The methods are effective, as we illustrate with examples. In the final section, which can be read independently of the others, we obtain more precise results for the Hilbert–Kunz function of the 3 variable power series zD−h(x,y).