Algebraic models of symmetric Nash sets

Abstract

The aim of this paper is to prove the existence of algebraic models for Nash sets having suitable symmetries. Given a Nash set $$M \subset {\mathbb {R}}^n$$ , we say that $$M$$ is specular if it is symmetric with respect to an affine subspace $$L$$ of $${\mathbb {R}}^n$$ and $$M \cap L=\emptyset $$ . If $$M$$ is symmetric with respect to a point of $${\mathbb {R}}^n$$ , we call $$M$$ centrally symmetric. We prove that every specular compact Nash set is Nash isomorphic to a specular real algebraic set and every specular noncompact Nash set is semialgebraically homeomorphic to a specular real algebraic set. The same is true replacing “specular” with “centrally symmetric”, provided the Nash set we consider is equal to the union of connected components of a real algebraic set. Less accurate results hold when such a union is symmetric with respect to a plane of positive dimension and it intersects that plane. The algebraic models for symmetric Nash sets $$M$$ we construct are symmetric. If the local semialgebraic dimension of $$M$$ is constant and positive, then we are able to prove that the set of birationally nonisomorphic symmetric algebraic models for $$M$$ has the power of continuum.