Almost regular quaternary quadratic forms
type: Item
instance of: scholarly article
title: Almost regular quaternary quadratic forms
zbMATH Open document ID: 1162.11020
author: Jacek Bochnak
author: Byeong-Kweon Oh
publication date: 2008-08-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10355
review text: Let us say that a quadratic form is integral if it is a positive definite nonsingular form with an integer coefficient matrix. An integral quadratic form \(f\) is called \textit{almost regular} (respectively, \textit{regular}) if it represents all sufficiently large (respectively, all) integers which are represented by \(f\) over \(p\)-adic integers \(\mathbb Z_p\) for every prime \(p\). The paper under review investigates the properties of almost regular quaternary quadratic forms. More specifically, a form is called \textit{\(p\)-isotropic} for a given prime \(p\) if it represents zero nontrivially over \(\mathbb Z_p\), and it is called \textit{\(p\)-anisotropic} otherwise. It is an old result contained in many of the classical papers on the subject that every positive definite integral quadratic form which is \(p\)-isotropic for every prime \(p\) is almost regular. An almost regular quadratic form is called \textit{exceptional} if it is \(p\)-anisotropic for at least one prime \(p\). It is known that there are no exceptional forms in more than four variables, and that all almost regular forms in less than four variables are exceptional. In this paper, the authors study exceptional quaternary forms. They show that although there are infinitely many such forms, in a certain sense they are rare. In particular, the authors prove that any exceptional quaternary form can be \(p\)-anisotropic for only one prime \(p\), and this prime has to be no greater than 37. On the other hand, they also show that for \textit{every} prime \(p \leq 37\), there exists an almost regular \(p\)-anisotropic quaternary form. To this end, for each prime \(p \leq 37\) the authors introduce the family \(\mathcal E_p\) of all primitive almost regular \(p\)-anisotropic quaternary quadratic forms, along with its subfamily \(\mathbb R_p\) of regular \(p\)-universal forms (if \(p >2\); \(2\mathbb Z_2\)-universal if \(p=2\)). The authors show that each \(\mathbb R_p\) is finite and define a subfamil…
reviewed by: Lenny Fukshansky
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 11E20
zbMATH DE Number: 5315502
zbMATH Keywords: quadratic equations
zbMATH Keywords: almost regular quadratic forms
MaRDI profile type: Publication
OpenAlex ID: W2315681114
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DOI: 10.5802/AIF.2391
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5315502