Semi-algebraic geometry with rational continuous functions

TL;DR

This paper explores the extension of semi-algebraic geometry by studying algebraically constructible functions and semi-algebraic subsets on affine real algebraic sets using rational continuous functions instead of polynomials.

Contribution

It introduces a framework for analyzing semi-algebraic sets and constructible functions with rational continuous functions, expanding classical polynomial-based methods.

Findings

Characterization of semi-algebraic subsets using rational continuous functions

Description of algebraically constructible functions in this new setting

Extension of classical semi-algebraic geometry results

Abstract

Let X be an affine real algebraic set . We investigate on the theory of algebraically constructible functions on X and the description of the semi-algebraic subsets of X when we replace the polynomial functions on X by some rational continuous functions on X.