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A quasianalytic class with weakly smooth germs

R. Guénet

TL;DR

The paper constructs an o-minimal expansion of the real field using weakly $C^ olinebreak[4]efty$ germs encoded by a function $H$ with a weakly $C^ olinebreak[4]efty$ germ that is not $C^ olinebreak[4]efty$. It develops a local-to-global framework based on $H$-basic sets, simple sub-$ ext{Λ}$-sets, and monomialization, culminating in a Fiber Cutting program that connects local parametrizations to a global Gabrielov-type complement result. The main contributions are the local simple-$ ext{Λ}$-set decomposition, the constant-rank parametrizations via $H$-manifolds, and the Local/Global Fiber Cutting Lemmas, which together establish model-completeness and o-minimality of $R_H$ without $C^ olinebreak[4]efty$-cell decomposition. This work provides a robust framework for o-minimality in quasianalytic and weakly smooth settings and points toward broader quantifier-elimination results under weaker hypotheses, with potential applications to generalized quasianalytic classes of weakly $C^ olinebreak[4]efty$ germs.

Abstract

In [O. Le Gal, J.-P. Rolin. An o-minimal structure which does not admit $C^\infty$ cellular decomposition. In: Ann. Inst. Fourier 59 (2009), pp 543-562], the authors construct an o-minimal structure which does not admit smooth cell-decomposition. We explain in an appendix that their proof of o-minimality is incomplete and we give a complete proof in the main text.

A quasianalytic class with weakly smooth germs

TL;DR

The paper constructs an o-minimal expansion of the real field using weakly germs encoded by a function with a weakly germ that is not . It develops a local-to-global framework based on -basic sets, simple sub--sets, and monomialization, culminating in a Fiber Cutting program that connects local parametrizations to a global Gabrielov-type complement result. The main contributions are the local simple--set decomposition, the constant-rank parametrizations via -manifolds, and the Local/Global Fiber Cutting Lemmas, which together establish model-completeness and o-minimality of without -cell decomposition. This work provides a robust framework for o-minimality in quasianalytic and weakly smooth settings and points toward broader quantifier-elimination results under weaker hypotheses, with potential applications to generalized quasianalytic classes of weakly germs.

Abstract

In [O. Le Gal, J.-P. Rolin. An o-minimal structure which does not admit cellular decomposition. In: Ann. Inst. Fourier 59 (2009), pp 543-562], the authors construct an o-minimal structure which does not admit smooth cell-decomposition. We explain in an appendix that their proof of o-minimality is incomplete and we give a complete proof in the main text.

Paper Structure

This paper contains 17 sections, 33 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A local property of $H$-basic sets
  3. Simple sub-$\Lambda$-sets
  4. The main result
  5. Composition and implicit functions
  6. Monomial division
  7. Parametrization results
  8. Monomialization
  9. A first parametrization
  10. A parametrization such that $\Pi_n$ has constant rank
  11. Local charts
  12. Cutting fibers
  13. Dimension
  14. A consequence of quasianalyticity
  15. The Local Fiber Cutting Lemma
  16. ...and 2 more sections

Key Result

Theorem 1.2

There exists a function $H \colon \mathbb{R} \to \mathbb{R}$ which satisfies the following conditions.

Figures (1)

  • Figure 1:

Theorems & Definitions (77)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof : Proof of Theorem \ref{['basic-simple-sub-lambda-set']}
  • Definition 2.6
  • Remark 2.7
  • ...and 67 more