$C^1$-triangulations of semialgebraic sets

TL;DR

This paper proves that every semialgebraic set can be triangulated with $C^1$ differentiable simplices, simplifying the theory of integration over such sets and extending results to all real closed fields.

Contribution

It introduces $C^1$-triangulations for semialgebraic sets and provides a new, simplified approach to defining integrals over these sets, applicable over all real closed fields.

Findings

Existence of $C^1$-triangulations for all semialgebraic sets.

Simplified definition of integration over semialgebraic sets.

Applicability over non-archimedean real closed fields.

Abstract

We show that every semialgebraic set admits a semialgebraic triangulation such that each closed simplex is $C^1$ differentiable. As an application, we give a straightforward definition of the integration $\int_X \omega$ over a compact semialgebraic subset $X$ of a differential form $\omega$ on an ambient algebraic manifold, that provides a significant simplification of the theory of semialgebraic singular chains and integrations. Our results hold over every (possibly non-archimedian) real closed field.