Hilbert-kunz Functions of 2 × 2 Determinantal Rings

Let k be an arbitrary field (of arbitrary characteristic) and let X = [xi,j] be a generic m × n matrix of variables. Denote by I2(X) the ideal in k[X] = k[xi,j ∶ i = 1, . . . ,m; j = 1, . . . , n] generated by the 2 × 2 minors of X. We give a recursive formulation for the lengths of the k[X]module k[X]/(I2(X) + (x q 1,1, . . . , x q m,n)) as q varies over all positive integers using Gröbner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We apply our method to give closed forms for these Hilbert-Kunz functions for cases m ≤ 2.