Explicit Hilbert-Kunz functions of \(2\times 2\) determinantal rings (Q2346534)

Let \((R,\mathfrak m, k)\) be a local or standard graded algebra with (graded) maximal ideal \(\mathfrak m\) of characteristic \(p\). The Hilbert-Kunz function of \(R\) is defined to be \(\mathrm{HK}_R(k) = \mathrm{ length } R/ \mathfrak m^{[p^k]}\). The Hilbert-Kunz multiplicity of \(R\) is defined by \[ e_{\mathrm{HK}}(R) = \lim_{q \to \infty} \frac{\mathrm{HK}_R(q)}{q^d} \] where \(d\) is the dimension of the ring. In general, it is a difficult problem to compute the Hilbert-Kunz multiplicity. Even in the case of minors of generic matrices, not much is known. In the paper under review, the authors consider the generalized Hilbert-Kunz function for two by two minors of generic matrices. More precisely, let \(R = k[x_{i,j}:i=1,\dots,m; j = 1, \dots, n]\) be the polynomialring in \(mn\) variables over a field \(k\) of arbitrary characteristic. The generalized Hilbert-Kunz function of an ideal \(I\) in \(R\) is defined to be \[ \mathrm{HK}_{R,I}(q) = \mathrm{ length } R/(I + (x_{1,1}^q,\dots, x_{m,n}^q)). \] When \(I = I_2(X)\), \textit{L. E. Miller} and \textit{I. Swanson} [Ill. J. Math. 57, No. 1, 251--277 (2013; Zbl 1308.13026)] gave a recursive formula for the generalized Hilbert-Kunz function. They gave a closed formula in the case \(m = 2\) and then deduce the formula for Hilbert-Kunz multiplicity. In the current paper, the authors extend the result by giving a closed formula for the general case. Precisely, in Theorem 3.3, the formula is given as \[ \sum_{i=1}^n (-1)^{n-i} \binom{n}{i} \binom{iq+m-1}{m+n-1} + \sum_{i=1}^n \sum_{j=1}^m (-1)^{m-j+i-1}\binom{n}{i}\binom{m}{j} \binom{jq-iq +n-1}{m+n-1}. \] Nevertheless, as the formula is expressed via sum of certain binomial coefficients, thus it is still not clear how to derive explicit formula for the Hilbert-Kunz multiplicity in these cases.