On the geometric and differential properties of closed sets definable in quasianalytic structures

TL;DR

This paper extends key geometric and differential properties known for subanalytic sets to those definable in quasianalytic structures, establishing equivalences among stratification, Chevalley estimates, and semicontinuity.

Contribution

It demonstrates that properties like stratification and semicontinuity, previously known for subanalytic sets, also hold for sets in quasianalytic o-minimal structures, broadening their applicability.

Findings

Uniform Chevalley estimate implies stratification by initial exponents

Stratification by initial exponents implies Zariski semicontinuity

Results hold for definable sets in quasianalytic structures

Abstract

In this paper we show that the equivalences between certain properties of closed subanalytic sets proved by E. Bierstone and P. Milman in \cite{[BM-1]} hold for closed sets definable in quasianalytic o-minimal structures. In particular we prove that uniform Chevalley estimate implies a stratification by the diagram of initial exponents and further, Zariski semicontinuity of the diagram of initial exponents. We also show that the stratification by the diagram implies Zariski semicontinuity of Hilbert-Samuel function.