Complex cells in sharply o-minimal structures

TL;DR

This paper extends the theory of complex cells to sharply o-minimal structures, providing effective cellular decomposition and parameterization theorems that are definable within these structures.

Contribution

It introduces a definable cellular preparation and parameterization framework for sharply o-minimal structures, generalizing previous work and ensuring polynomial effectiveness.

Findings

Cellular decomposition theorems are polynomially effective.

Constructions are definable within the structures.

Applicable to sets in reducts of R_an, yielding definable cells and maps.

Abstract

We extend the theory of complex cells introduced by Binyamini and Novikov to the sharply o-minimal setting, obtaining cellular preparation and parameterization theorems which are polynomially effective in the degrees of the relevant sets. Our constructions are definable, and so applying them to sets in a given reduct of R_an yields cells and cellular maps definable in the same reduct.