Hilbert-Kunz function of monomial ideals and binomial hypersurfaces (Q1817355)

Let \((R,m)\) be a local ring or standard graded \(K\)-algebra. If \(m=Rx_1 + \cdots +Rx_n\), the Hilbert-Kunz function \(HK\) of \(R\) with respect to \(x_1, \dots, x_n\) is defined, for \(q\geq 1\), by \(HK(q) =\ell (R/x^{[q]})\) where \(x^{[q]} =Rx^q_1+ \cdots+ Rx^q_n\). In summary the two main results are: (1) Let \(R=S/I\) with \(I\) a monomial ideal of \(S= K[X_1, \dots, X_n]\). Then, for large \(q\), \(HK(q) =P(q)\) where \(P(Y)\) is a polynomial of degree \(\dim R\) and has as leading coefficient the multiplicity \(e(R)\). (2) Let \(R=S/I\) where \(I\) is the principal ideal generated by a homogeneous binomial form \(X^a-X^b\) with \(\text{gcd} (X^a,X^b) =1\). Then \(HK(q) =P(q, \varepsilon)\) for large \(q\), where the polynomial \(P(Y,Z)\) is explicitly determined and \(\varepsilon\) is \(q\) mod the highest component in the vectors \(a\) and \(b\).