On primitively universal quadratic forms
type: Item
instance of: scholarly article
title: On primitively universal quadratic forms
zbMATH Open document ID: 1247.11047
published in: Lithuanian Mathematical Journal
publication date: 2011-01-24T00:00:00Z
review text: If a positive definite integral quadratic form \(Q\) in four or more variables is primitively \(p\)-universal for all primes \(p\) (that is, \(Q\) primitively represents all positive integers over each of the rings \(\mathbb Z_p\) of \(p\)-adic integers), then it is known that \(Q\) primitively represents all sufficiently large positive integers (see \textit{J. W. S. Cassels} [Rational quadratic forms. London, New York, San Francisco: Academic Press (1978; Zbl 0395.10029)]). In this paper, the author studies conditions on a \(p\)-adic Jordan decomposition of \(Q\) that guarantee that \(Q\) is primitively \(p\)-universal. The case of an odd prime \(p\) can be resolved using standard facts about the representation properties of unimodular lattices over \(\mathbb Z_p\). The case \(p=2\) is predictably more complicated, and the author obtains only incomplete results in this case. The local results are applied to obtain the following global result: If \(Q\) is a positive definite classically integral quadratic form of rank at least four with odd square-free determinant, and \(Q\) primitively represents the integers \(1\), \(4\) and \(8\), then \(Q\) primitively represents all sufficiently large positive integers.
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 5840703
zbMATH Keywords: almost primitively universal quadratic forms
zbMATH Keywords: primitively p-universal quadratic forms
author: Natalia V. Budarina
reviewed by: Andrew G. Earnest
MaRDI profile type: Publication
full work available at URL: s10986-010-9076-2
OpenAlex ID: W2031951198
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DOI: 10.1007/S10986-010-9076-2
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description: scientific article; zbMATH DE number 5840703