COMPUTING HILBERT–KUNZ FUNCTIONS OF 1-DIMENSIONAL GRADED RINGS

Abstract. According to a theorem of Monsky, the Hilbert–Kunz function of a 1-dimensional standard graded algebra R over a finite field K has, for i 0, the shape HKR(i) = c(R) · p i + ϕ(i), where c(R) is the multiplicity of R and ϕ is a periodic function. Here we study explicit computer algebra algorithms for computing such Hilbert–Kunz functions: the period length and the values of ϕ, as well as a concrete number N ≥ 0 such that the description above holds for i ≥ N. 1. Introduction. In his papers [7] and [8] E. Kunz introduced and stud-ied the function i 7 → `(R/m[pi]) for a noetherian local ring (R,m) of charac-teristic p, where m[p i] denotes the ith Frobenius power of m. He called it the length of the Frobenius fibers. Later, in [6], P. Monsky called it the Hilbert