On spectral types of semialgebraic sets

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 ${\mathcal{S}}^{\ast }(M)$ 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 ${\mathcal{S}}^{\ast }(M)$ are isomorphic if and only if $M$ is compact. On the other hand, their respective maximal spectra ${\beta }_{s}M$ and ${\beta }_s^{\ast }M$ 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 $\partial M:={\beta }_s^{\ast }M\setminus M$ associated with formal paths.

22 pages

References in corpus (2)

• On the Krull dimension of rings of semialgebraic functions
• On the substitution theorem for rings of semialgebraic functions