On spectral types of semialgebraic sets

TL;DR

This paper shows that semialgebraic sets are uniquely characterized by their rings of continuous semialgebraic functions, with distinctions based on boundedness and compactness, and explores the topological properties of their spectra.

Contribution

It establishes the precise relationship between semialgebraic sets and their function rings, including conditions for isomorphism and homeomorphism of spectra.

Findings

Semialgebraic sets are determined by their rings of continuous semialgebraic functions.

The rings ${ m S}(M)$ and ${ m S}^*(M)$ are isomorphic iff $M$ is compact.

Spectra $eta_s M$ and $eta_s^* M$ are always homeomorphic, classifying a large part of $M$.

Abstract

In this work we prove that a semialgebraic set $M\subset{\mathbb R}^m$ is determined (up to a semialgebraic homeomorphism) by its ring ${\mathcal S}(M)$ of (continuous) semialgebraic functions while its ring ${\mathcal S}^*(M)$ of (continuous) bounded semialgebraic functions only determines $M$ besides a distinguished finite subset $\eta(M)\subset M$. In addition it holds that the rings ${\mathcal S}(M)$ and ${\mathcal S}^*(M)$ are isomorphic if and only if $M$ is compact. On the other hand, their respective maximal spectra $\beta_s M$ and $\beta_s^* M$ endowed with the Zariski topology are always homeomorphic and topologically classify a `large piece' of $M$. The proof of this fact requires a careful analysis of the points of the remainder $\partial M:=\beta_s^* M\setminus M$ associated with formal paths.