The membership problem for finitely generated quadratic modules in the univariate case
type: Item
instance of: scholarly article
title: The membership problem for finitely generated quadratic modules in the univariate case
zbMATH Open document ID: 1273.12002
author: Doris Augustin
published in: Journal of Pure and Applied Algebra
publication date: 2013-01-30T00:00:00Z
review text: A quadratic module of a ring \(R\) is a subset \(Q \subseteq R\) that contains \(1\) and is closed under addition and under multiplication with squares of elements of \(R\). If moreover \(Q\) is closed under multiplication, it is called a preordering. In the case \(R = \mathbb{R}[X]\) one associates the semi-algebraic set \(S := \{x \in \mathbb R \mid g(x) \geq 0\,\, \forall g \in Q\}\) to \(Q\). The central result of the present article is a complete classification of all finitely generated quadratic modules \(Q\) for which the set \(S\) is compact (Theorem~3.2 and Corollary~4.5). Note that in that case, \(Q\) is automatically closed under multiplication, i.e., it is a preordering. This classification implies that \(Q\) is first-order definable in a suitable sense. In the case where \(S\) is not compact, definability was already known before, so one now has definability for any finitely generated quadratic module \(Q\) of \(\mathbb{R}[X]\) (Theorem~3.3). From this, the author deduces that the membership problem for such a \(Q\) is solvable, i.e., there exists an algorithm deciding whether a given polynomial lies in \(Q\) (Corollary~3.4). As another application of the classification result, the author obtains a natural set of generators for any finitely generated quadratic module whose set \(S\) is compact. This generalizes results of \textit{S. Kuhlmann} and \textit{M. Marshall} [Trans. Am. Math. Soc. 354, No.11, 4285--4301 (2002; Zbl 1012.14019)].
reviewed by: Immanuel Halupczok
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 6132788
zbMATH Keywords: Quadratic module
zbMATH Keywords: preordering
zbMATH Keywords: membership problem
zbMATH Keywords: natural generators
MaRDI profile type: Publication
full work available at URL: j.jpaa.2012.02.004
OpenAlex ID: W2038527060
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DOI: 10.1016/J.JPAA.2012.02.004
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description: scientific article; zbMATH DE number 6132788