iv : 0 90 7 . 24 70 v 1 [ m at h . A C ] 1 5 Ju l 2 00 9 Algebraicity of some Hilbert - Kunz multiplicities ( modulo a conjecture )

Let F be a finite field of characteristic 2 and h be the element x3 + y3 + xyz of F [[x, y, z]]. In an earlier paper we made a precise conjecture as to the values of the colengths of the ideals (x, y, z, h) for q a power of 2. We also showed that if the conjecture holds then the Hilbert-Kunz series of H = uv+ h is algebraic (of degree 2) over Q(w), and that μ(h) is algebraic (explicitly, 4 3+ 5 14 √ 7 ). In this note, assuming the same conjecture, we use a theory of infinite matrices to rederive this result, and we extend it to a wider class of H; for example H = g(u, v) + h. In a follow-up paper, under the same hypothesis, we will show that transcendental Hilbert-Kunz multiplicities exist.