EXPLICIT HILBERT-KUNZ FUNCTIONS OF 2 x 2 DETERMINANTAL RINGS

摘要

Let k[X] = k[x(i,j) : i = 1, ... , m; j = 1, ... , n] be the polynomial ring in mn variables x(i,j) over a field k of arbitrary characteristic. Denote by I-2(X) the ideal generated by the 2 x 2 minors of the generic m x n matrix [x(i,j)]. We give a closed polynomial formulation for the dimensions of the k-vector space k[X]/(I-2(X) + (x(1.1)(q), ... , x(m,n)(q))) as q varies over all positive integers, i.e., we give a closed polynomial form for the generalized Hilbert-Kunz function of the determinantal ring k[X]/I-2(X). We also give a closed formulation of dimensions of other related quotients of k[X]/I-2(X). In the process we establish a formula for the numbers of some compositions (ordered partitions of integers), and we give a proof of a new binomial identity.