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Complex cells in sharply o-minimal structures

Gal Binyamini, Oded Carmon, Dmitry Novikov

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.

Complex cells in sharply o-minimal structures

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.

Paper Structure

This paper contains 23 sections, 19 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Sharply o-minimal structures
  5. Complex cells
  6. Cells and cellular maps
  7. Extensions and cellular covers
  8. Sharp cellular parameterization
  9. Outline of the proof of the #CPT
  10. Removing an ε-net
  11. Radial bound
  12. Proof of the fundamental lemma
  13. Proofs of the #CPT and #CPrT
  14. tocStep 1: $\text{# CPT}_1$tocStep 1
  15. Proper projections near a puncture
  16. ...and 8 more sections

Key Result

Theorem A

Let $\mathcal{C}=\mathcal{F}_1\odot\cdots\odot\mathcal{F}_\ell\subset\mathbb{C}^\ell$ be a complex cell and let $Z_1,\dots,Z_k$ be a finite collection of analytic hypersurfaces of $\mathcal{C}$, all of them definable in an o-minimal structure $\mathcal{S}$. Then there is a finite collection of compl

Figures (2)

  • Figure 1: Removing an $\varepsilon$-net. By inductively applying the # CPT in the base, we reduce to the case where the projection of $Z$ to the last coordinate does not intersect the set $\varepsilon\mathbb{Z}^2\cap A\qty(1,6)$.
  • Figure 2: Bounding $\operatorname{diam}\qty(\operatorname{abs}({\pi_{\ell+1}(Z_0)}) ; \mathbb{R})$ when the type of $\mathcal{C}$ contains annuli. All change in the radial direction of $\pi_{\ell+1}(Z_0)$ is accounted for by either a connected component $\Gamma_i$ of the boundary of $\pi_{\ell+1}(Z_0)$ or by an annulus $\Gamma^*_i$ which is entirely contained in $\pi_{\ell+1}(Z_0)$. The boundary of $\pi_{\ell+1}(Z_0)$ has few connected components, each of which has small radial diameter by the case where $\mathcal{C}$ is a cellular polydisc. There are also few such maximal annuli $\Gamma^*_i$ and, since they cannot contain any point of an $\varepsilon$-net, their width is small.

Theorems & Definitions (61)

  • Theorem A: Sharp Cellular Parametrization Theorem, # CPT
  • Theorem B: Sharp Cellular Preparation Theorem, # CPrT
  • Remark B: Asymptotic notation
  • Definition B
  • Definition B
  • Definition B
  • Definition B
  • Proposition B: BinyaminiNovikovZak2022
  • Proposition B: BinyaminiNovikovZak2022
  • Example B
  • ...and 51 more