Explicit Hilbert-Kunz functions of 2 x 2 determinantal rings

الأصل · EN

Let k[X] = k[xᵢ,ⱼ: i = 1,..., m; j = 1,..., n] be the polynomial ring in m n variables xᵢ,ⱼ over a field k of arbitrary characteristic. Denote by I₂(X) the ideal generated by the 2 × 2 minors of the generic m × n matrix [xᵢ,ⱼ]. We give a closed formulation for the dimensions of the k-vector space k[X]/(I₂(X) + (x₁,₁q,..., xₘ,ₙq)) as q varies over all positive integers, i.e., we give a closed form for the generalized Hilbert-Kunz function of the determinantal ring k[X]/I₂[X]. We also give a closed formulation of dimensions of related quotients of k[X]/I₂[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.