The Derivative of a Constructible Function is Constructible

TL;DR

This paper proves that the class of constructible functions in tame real geometry remains stable when derivatives are taken, extending the understanding of their structural properties.

Contribution

It establishes that the derivative of a constructible function is also constructible, a new result in the theory of tame geometry and parametric integration.

Findings

Constructible functions are stable under differentiation.

The class of globally subanalytic functions is closed under derivatives.

This stability aids in parametric integration and tame geometry analysis.

Abstract

The notion of constructible functions in the setting of tame real geometry has been introduced by Cluckers and Dan Miller in their work on parametric integration of globally subanalytic functions. A function on a globally subanalytic set is called constructible if it is a finite sum of finite products of globally subanalytic functions and the logarithm of positive globally subanalytic functions. We show that the class of constructible functions is stable under taking derivatives.