Some surprising Hilbert-Kunz functions
This article determines the ``Hilbert-Kunz'' function \(n \to e_ n\) of the \(a\)-dimensional local ring \({\mathfrak O}=\mathbb{Z}/p[[X_ 1, \dots,X_{a+1}]]/(\sum^{a+1}_ 1X_ i^{d_ i})\); \((e_ n\) is the \(\mathbb{Z}/p\)-dimension of \({\mathfrak O}/(X^{p^ n}_ 1 {\mathfrak O}, \dots, X^{p^ n}_{a+1} {\mathfrak O}))\). It is known that \(e_ n=Cp^{an}+\Delta_ n\) where \(C \geq 1\) and \(\Delta_ n=O (p^{(a- 1)n})\). The following are proved: (1) \(C\) is rational; (2) \(\Delta_ n=O(p^{(a-2) n})\); (3) If \(a=2\) or \(p=2\), \(\Delta_ n\) is an eventually periodic function of \(n\); (4) If \(a>2\) and \(p>2\) there are integers \(\ell^ \#\) and \(\mu\), \(\mu \geq 1\), such that \(\Delta_{n+\mu} = \ell^ \#\). \(\Delta_ n\) for \(n \gg 0\). For example when \(p=3\), \(a=4\) and each \(d_ i=2\) it is shown that \(\mu+1\), \(\ell^ \#=5\) and \(e_ n=(23/19)\) \((81)^{n}-(4/19)\) \((5)^ n\) for all \(n\). The article extends the thesis of the first author in which (3) was proved: the new ingredient is the study of a very natural representation ring.
scientific article; zbMATH DE number 561290