About certain subsets of the Euclidean space
Let \(\tilde{\mathcal A}_ m\) be the algebra of functions on \(\mathbb{R}^ n\) generated by polynomial functions and exponentials of linear forms. The subset \(S\) in \(\mathbb{R}^ n\) belongs to \({\mathcal P}_ n\) if and only if there exist \(m\) and \(F\) in \(\tilde{\mathcal A}_{n+m}\) for which \(S\) is the image of the zerosubset of \(F\) by the canonical projection of \(\mathbb{R}^{n+m}\) onto \(\mathbb{R}^ n\). Let \(\tilde{\mathcal P}_ n\) be the smallest subset of parts in \(\mathbb{R}^ n\) which contains \({\mathcal P}_ n\), their closures and the images by the canonical projection of the elements in \(\tilde{\mathcal P}_{n+m}\). --- This family of sets is defined by an induction in two steps. The main goal of this article is to prove that \(\tilde{\mathcal P}_{n+m}\) contains the complementary part of each element in \(\tilde{\mathcal P}_{n+m}\), the union and the intersection of every finite family in \(\tilde{\mathcal P}_{n+m}\). The key results for the proof are theorems by A. G. Khovanskij on the set of the solutions of a Pfaff system on a Pfaff manifold.
scientific article; zbMATH DE number 55932