On Hilbert-Kunz functions of some hypersurfaces
Let \((A,\mathfrak m)\) denote a complete local \(\mathbb Z/(p)\)-algebra, where \(p\) denotes a prime number. Let \(e_n(A)=\text{length}(A/\mathfrak m_n),\) where \(\mathfrak m_n\) is the ideal of \(A\) generated by all \(a^q\), \(q=p^n\), with \(a\in\mathfrak m\), denote the (local) Hilbert-Kunz function. It was shown by \textit{P. Monsky} in Math. Ann. 263, 43-49 (1983; Zbl 0509.13023) that \(e_n(A)=cp^{dn}+\Delta_n\), where \(d=\dim A\) and \(\Delta_n=O(p^{n(d-1)})\). Put \(B=\mathbb Z/(p)X_1, \ldots, X_s\) and \(A=B/fB\). In the case of \(f=X_1^{d_1}-\prod_{i=2}^s X_i^{d_i}\) (resp. \(f = \sum_{i=1}^s X_i^{d_i}\)) it was shown by \textit{E. Kunz} in Am. J. Math. 98, 999-1013 (1976; Zbl 0341.13009), resp. by \textit{C. Han} and \textit{P. Monsky} in Math. Z. 214, 119-135 (1993; Zbl 0788.13008) that \(c\) is rational and \(\Delta_n\) is eventually periodic. In the main result of the paper under review this is generalized to the case of \(f\) a certain sum of powers of squarefree monomials. As a main technical tool the authors use the representation ring developed by C. Han and P. Monsky (loc. cit.). Moreover this generalization covers also the a result of \textit{A. Conca} shown in Manuscr. Math. 90, No. 3, 287-300 (1996; Zbl 0882.13019).
scientific article; zbMATH DE number 1117779