UPC condition with parameter for subanalytic sets

TL;DR

This paper studies the UPC property of sections of relatively compact open subanalytic sets, showing that two parameters can be chosen independently of the section while the third depends on the specific point.

Contribution

It extends the understanding of UPC conditions by analyzing parameter dependencies for sections of subanalytic sets.

Findings

Two of the three UPC parameters are independent of the section.

The third UPC parameter generally depends on the specific section point.

The results generalize previous work on subanalytic sets.

Abstract

In 1986 Paw{\l}ucki and Ple\'sniak introduced the notion of {\sl uniformly polynomially cuspidal} (UPC) sets and proved that every relatively compact and fat subanalytic subset of ${\Rz}^n$ satisfies the UPC condition. Herein we investigate the UPC property of the sections of a relatively compact open subanalytic set $E\subset{\Rz}^k\times{\Rz}^n$ and we show that two of the three parameters in the UPC condition can be chosen independently of the section, while the third one depends generally on the point defining the section.