Real Algebraic and Analytic Geometry |

Algebraic models of symmetric Nash sets.

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Submission: 2012, August 28.

*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 wrt an affine subspace $L$ of $\mathbb{R}^n$
and $M \cap L=\emptyset$. If $M$ is symmetric wrt 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 wrt 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
.*

Mathematics Subject Classification (2010): 14P05, 14P20, 14P25.

Keywords and Phrases: Algebraic models of Nash sets, symmetric Nash sets, connected components of real algebraic sets.

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