Rationality of Hilbert-Kunz multiplicities: a likely counterexample
type: Item
instance of: scholarly article
title: Rationality of Hilbert-Kunz multiplicities: a likely counterexample
zbMATH Open document ID: 1188.13007
published in: Michigan Mathematical Journal
publication date: 2009-09-17T00:00:00Z
review text: The Hilbert-Kunz multiplicity of a hypersurface is in many cases a rational number, but the author believes that \(F=uv+x^3+y^3+xyz\in {\mathbb Z}_2[x,y,z,u,v]\) has the Hilbert-Kunz multiplicity \((4/3)+(5/14\sqrt{7})\). Let \(e_{nj}\) be the colength of the ideal \((x^q,y^q,z^q,(x^3+y^3+xyz)^j)\subset {\mathbb Z}_2[x,y,z]\), where \(q=2^n\), \(n\in {\mathbb N}\). Then \(e_{n1}\) is \((7q^2-q-3)/3\) or \((7q^2-q-5)/3\) according as \(q\equiv 1\) or \(2\) mod \(3\) [see \textit{R.-O. Buchweitz} and \textit{Q. Chen}, J. Algebra 197, No. 1, 246--267 (1997; Zbl 0898.13009)]. Here it is conjectured a formula for \(e_{nj}\), which implies the above non rational value of the Hilbert-Kunz multiplicity of \(F\).
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 14G17
zbMATH DE Number: 5604550
zbMATH Keywords: Hilbert-Kunz multiplicity
zbMATH Keywords: colength of an ideal
zbMATH Keywords: hypersurface
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1220879427
OpenAlex ID: W2076079599
Wikidata QID: Q124854314
cites work: Hilbert-Kunz functions of cubic curves and surfaces
cites work: \(p\)-fractals and power series. I: Some 2 variable results
cites work: \(P\)-fractals and power series. II: Some applications to Hilbert-Kunz theory
DOI: 10.1307/MMJ/1220879427
author: Paul Monsky
description: scientific article; zbMATH DE number 5604550