Locally injective semialgebraic maps
E. Baro, J. F. Fernando, J. M. Gamboa
TL;DR
This paper forges a precise link between local injectivity of semialgebraic maps and finiteness properties of the induced homomorphism between rings of semialgebraic functions. By analyzing the finiteness notions (finite, integral, simple, finitely generated) for $\varphi_{\pi}:{\mathcal{S}}(N)\to{\mathcal{S}}(M)$, it shows that $\varphi_{\pi}$ being finite implies that the spectral map $\operatorname{Spec_s}(\pi)$ and the map $\pi$ are proper, separated, and have finite fibers, with the additional structure that $\operatorname{β_s}\pi$ is well-behaved. In the compact case, the work yields a clean criterion: $\pi$ is locally injective if and only if $\varphi_{\pi}$ is finite, tying a geometric property directly to an algebraic finiteness condition. The results provide a robust bridge between semialgebraic geometry and the algebra of rings of semialgebraic functions, with implications for how local injectivity can be detected via ring-theoretic data.
Abstract
We characterize locally injective semialgebraic maps between two semialgebraic sets in terms of the induced homomorphism between their rings of (continuous) semialgebraic functions.