Exact values of the function \(\varGamma ^{*}(k)\)
type: Item
instance of: scholarly article
title: Exact values of the function \(\varGamma ^{*}(k)\)
zbMATH Open document ID: 1241.11040
author: Michael P. Knapp
published in: Journal of Number Theory
publication date: 2011-07-22T00:00:00Z
review text: The function \(\Gamma^*(k)\) is the least integer \(s\) such that any diagonal form of degree \(k\) over \(\mathbb{Q}\) in \(s\) variables has a non-trivial \(p\)-adic zero, for every prime \(p\). \textit{H. Davenport} and \textit{D. J. Lewis} [Proc. R. Soc. Lond., Ser, A 274, 443--460 (1963; Zbl 0118.28002)] showed that \(\Gamma^*(k)\leq k^2+1\) for all \(k\geq 1\), with equality when \(k+1\) is prime. In principle there is a standard finite computation to calculate \(\Gamma^*(k)\) for any individual \(k\), but until recently only the values for \(k\leq 11\), together with those for which \(k+1\) is prime, had been handled. The author, in work to appear, has improved the situation by treating odd \(k\leq 25\). The present paper evaluates \(\Gamma^*(k)\) for all \(k\leq 31\), and deals with the degree 32 case for all \(p\)-adic fields with \(p\not=2\). The proofs handle the various values of \(k\) separately, using a variety of techniques to reduce as much as possible the amount of computational testing required.
reviewed by: D. R. Heath-Brown
Mathematics Subject Classification ID: 11D88
Mathematics Subject Classification ID: 11D72
zbMATH DE Number: 5931163
zbMATH Keywords: Artin's conjecture
zbMATH Keywords: diagonal forms
zbMATH Keywords: calculations
zbMATH Keywords: \(p\)-adic fields
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.04.006
OpenAlex ID: W1592778625
cites work: Q5522902
cites work: Γ*(8)
cites work: Q3291629
cites work: Homogeneous additive equations
cites work: Homogeneous additive congruences
cites work: Cubic congruences
cites work: On a problem of Chowla and Shimura
DOI: 10.1016/J.JNT.2011.04.006
description: scientific article; zbMATH DE number 5931163