Sums of values represented by a quadratic form
type: Item
instance of: scholarly article
title: Sums of values represented by a quadratic form
zbMATH Open document ID: 1336.11027
author: Grégory Berhuy
author: Nicolas Grenier-Boley
author: M. G. Mahmoudi
published in: Manuscripta Mathematica
publication date: 2013-03-18T00:00:00Z
review text: Let \(K\) be a field of characteristic not \(2\) and let \(q\) be a nondegenerate quadratic form over \(K\) defined on a finite-dimensional \(K\)-vector space \(V\). The \(q\)-level \(s_q(K)\) (resp. \(q\)-sublevel \(\underline{s}_q(K)\)) is defined to be the smallest \(n\) for which there exist \(v_1,\ldots,v_n\in V\) (resp. \(v_1,\ldots,v_{n+1}\in V\setminus\{ 0\}\)) such that \(-1=q(v_1)+\cdots +q(v_n)\) (resp. \(0=q(v_1)+\cdots +q(v_{n+1})\)). If no such representation exists, the respective value is defined to be \(\infty\). If \(q\) is given by the diagonalized form \(\langle 1\rangle\), then the \(q\)-(sub)level is nothing else but the usual level \(s\) (resp. sublevel \(\underline{s}\)), in which case it is easy to see that \(s=\underline{s}\). A famous result by \textit{A. Pfister} [J. Lond. Math. Soc. 40, 159--165 (1965; Zbl 0131.25002)] states that the level, if finite, is always a \(2\)-power, and that each \(2\)-power can be realized as level of a suitably constructed field. The main purpose of the present paper is to extend these results to \(q\)-(sub)levels and to find upper bounds in terms of other field invariants pertaining to quadratic forms such as the \(u\)-invariant and the Pythagoras number. One first notes that if \(q\) is isotropic, then \(\underline{s}_q(K)=0\) and \(s_q(K)=1\), so that one may assume \(q\) to be anisotropic, in which case \(s_q(K)\leq \underline{s}_q(K)+1\), and \(\underline{s}_q(K)\leq s_q(K)\) if \(q\) represents \(1\). It is shown that to each positive integer \(n\) there exists a field \(K\) and a form \(q\) over \(K\) such that \(s_q(K)=n\). However, if \(q\) is an anisotropic Pfister form, or, more generally, a round form, i.e., a form where the represented values of \(q\) are exactly the similarity factors of \(q\), then \(s_q(K)=\underline{s}_q(K)=2^m\) for some nonnegative integer \(m\). If \(\dim(q)=3\), then, for any nonnegative integer \(k\), both \((2^{2k}+2)/3\) and \((2^{2k+1}+1)/3\) can be realized as \(s_q…
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E10
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 12D15
zbMATH DE Number: 6145962
zbMATH Keywords: quadratic form
zbMATH Keywords: level
zbMATH Keywords: sublevel
zbMATH Keywords: Pythagoras number
zbMATH Keywords: Pfister form
zbMATH Keywords: round form
zbMATH Keywords: formally real field
reviewed by: Detlev Hoffmann
MaRDI profile type: Publication
full work available at URL: s00229-012-0551-4
OpenAlex ID: W1982683089
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DOI: 10.1007/S00229-012-0551-4
description: scientific article; zbMATH DE number 6145962