The Hilbert-Kunz function attached to a characteristic p power series

Let k be a field of characteristic p, and F be a non-zero element of the maximal ideal of k[[x₀,...,x_s]]. If q=pⁿ, e_n will denote the colength of the ideal generated by F and the (x_i)^q. When s=1 it's known that

I'll concentrate on the (still mostly obscure) case s=2. Diverse methods, (stability theory for rank 2 bundles, p-fractals), show that

for certain classes of F.

When s>2 there are surprises, and remarkable examples (and conjectured examples) connected with the theory of p-fractals developed by Teixeira and me; I'll discuss these too.