The membership problem for finitely generated quadratic modules in the univariate case
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)].
scientific article; zbMATH DE number 6132788
scientific article