Morse functions definable in d-minimal structures

TL;DR

This paper proves that in d-minimal structures, the set of definable Morse functions is dense within the space of all definable smooth functions on a manifold, highlighting their abundance.

Contribution

It establishes the density of definable Morse functions in the space of all definable smooth functions in d-minimal structures.

Findings

Definable Morse functions are dense in the space of definable C^p functions.

The result applies to any definable C^p submanifold within a d-minimal expansion.

The paper advances understanding of the structure of definable functions in o-minimal-like settings.

Abstract

Fix a d-minimal expansion of an ordered field. We consider the space $\mathcal D^p(M)$ of definable $\mathcal C^p$ functions defined on a definable $\mathcal C^p$ submanifold $M$ equipped with definable $\mathcal C^p$ topology. The set of definable $\mathcal C^p$ Morse functions is dense in $\mathcal D^p(M)$.