A generalized Sard theorem on real closed fields

TL;DR

This paper extends Sard's theorem to semi-algebraic functions over arbitrary real closed fields, broadening the theorem's applicability beyond real numbers.

Contribution

It introduces a generalized notion of critical values and proves a Sard type theorem in the context of real closed fields.

Findings

Established a Sard type theorem for semi-algebraic functions on real closed fields

Generalized the concept of critical values in this broader setting

Extended classical results to a more general algebraic framework

Abstract

We work with semi-algebraic functions on arbitrary real closed fields. We generalize the notion of critical values and prove a Sard type theorem in our framework.