Stability of closedness of semi-algebraic sets under continuous semi-algebraic mappings

TL;DR

This paper proves that for most linear perturbations, the image of a closed semi-algebraic set under a continuous semi-algebraic map remains closed, by analyzing tangent cones at infinity and exceptional directions.

Contribution

It establishes the stability of closedness of semi-algebraic sets under generic linear perturbations using tangent cone analysis.

Findings

Generic linear perturbations preserve closedness of semi-algebraic images.

The set of exceptional directions at infinity is nowhere dense.

The approach involves tangent cones at infinity and exceptional directions.

Abstract

Given a closed semi-algebraic set $X \subset \mathbb{R}^n$ and a continuous semi-algebraic mapping $G \colon X \to \mathbb{R}^m,$ it will be shown that there exists an open dense semi-algebraic subset $\mathscr{U}$ of $L(\mathbb{R}^n, \mathbb{R}^m),$ the space of all linear mappings from $\mathbb{R}^n$ to $\mathbb{R}^m,$ such that for all $F \in \mathscr{U},$ the image $(F + G)(X)$ is a closed (semi-algebraic) set in $\mathbb{R}^m.$ To do this, we study the tangent cone at infinity $C_\infty X$ and the set $E_\infty X \subset C_\infty X$ of (unit) exceptional directions at infinity of $X.$ Specifically we show that the set $E_\infty X$ is nowhere dense in $C_\infty X \cap \mathbb{S}^{n - 1}.$