Clustered Cell Decomposition in P-Minimal Structures

TL;DR

This paper extends cell decomposition results in P-minimal structures by introducing regular clustered cells, which generalize classical cells without requiring definable Skolem functions, thus broadening the scope of geometric analysis in such structures.

Contribution

It proves that any definable set in a P-minimal structure can be partitioned into classical and regular clustered cells, generalizing prior results and removing the need for definable Skolem functions.

Findings

Definable sets can be partitioned into classical and clustered cells.

Clustered cells have similar structure to classical cells but lack a definable center.

The result broadens the applicability of cell decomposition in P-minimal structures.

Abstract

We prove that in a $P$-minimal structure, every definable set can be partitioned as a finite union of classical cells and regular clustered cells. This is a generalization of previously known cell decomposition results by Denef and Mourgues, which were dependent on the existence of definable Skolem functions. Clustered cells have the same geometric structure as classical, Denef-type cells, but do not have a definable function as center. Instead, the center is given by a definable set whose fibers are finite unions of balls.