On the one dimensional polynomial and regular images of Rn\R^n

In this work we present a full geometric characterization of the 1-dimensional polynomial and regular images $S$ of $\R^n$ and we compute for all of them the invariants ${\rm p(S)}$ and ${\rm r}(S)$, already introduced in \cite{fg2}

On the complements of 3-dimensional convex polyhedra as polynomial images of R3{\mathbb R}^3

We prove that the complement ${\mathcal S}:={\mathbb R}^3\setminus{\mathcal K}$ of a 3-dimensional convex polyhedron ${\mathcal K}\subset{\mathbb R}^3$ and its closure $\overline{{\mathcal S}}$ are polynomial images of ${\mathbb R}^3$. The former techniques cannot be extended in general to represent such semialgebraic sets ${\mathcal S}$ and $\overline{{\mathcal S}}$ as polynomial images of ${\mathbb R}^n$ if $n\geq4$.Comment: 12 pages, 1 figur

On spectral types of semialgebraic sets

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.Comment: 22 page