Similarity of quadratic forms and half-neighbors
type: Item
instance of: scholarly article
title: Similarity of quadratic forms and half-neighbors
zbMATH Open document ID: 0922.11031
author: Detlev Hoffmann
published in: Journal of Algebra
publication date: 1999-10-03T00:00:00Z
review text: The quadratic forms \(\phi\) and \(\psi\) of dimension \(2^n\) are called half-neighbors iff \(\phi \perp -a \psi\) is similar to an \((n + 1)\)-fold Pfister form for some scalar \(a\). If \(n \leq 2\) then every half-neighbors are similar but for \(n \geq 3\) counterexamples have been given by Izhboldin and by the author. So the aim of the paper under review is to explain under which circumstances half-neighbors of dimension \(2^n\) are similar for \(n \geq 3\). The results are: If \(\widetilde u(F) < 2^n\), \(n-2 \geq m \geq 1\) and \(I_t^{m + 1} F = 0\) then if one of the half-neighbors is congruent to an \(m\)-fold Pfister form modulo \(I^{m + 1}F\) then they are similar. If \(n \geq m \geq 2\) and \(F\) is \(m\)-linked, then every half-neighbors in \(I^mF\) of dimension \(2^n\) are similar. Moreover, the author discusses dependencies between various equivalence relations for quadratic forms: similarity, birational equivalence, stable birational equivalence and half-neighbor equivalence. Some partial results are here established.
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 1218696
zbMATH Keywords: quadratic form
zbMATH Keywords: Pfister neighbor
zbMATH Keywords: similarity
reviewed by: Krzysztof Kozioł
MaRDI profile type: Publication
full work available at URL: jabr.1997.7316
OpenAlex ID: W2063459054
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DOI: 10.1006/JABR.1997.7316
description: scientific article; zbMATH DE number 1218696