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Quantitative approximate definable choices

Antonio Lerario, Luca Rizzi, Daniele Tiberio

Abstract

In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite-dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest.

Quantitative approximate definable choices

Abstract

In semialgebraic geometry, projections play a prominent role. A definable choice is a semialgebraic selection of one point in every fiber of a projection. Definable choices exist by semialgebraic triviality, but their complexity depends exponentially on the number of variables. By allowing the selection to be approximate (in the Hausdorff sense), we improve on this result. In particular, we construct an approximate selection whose degree is linear in the complexity of the projection and does not depend on the number of variables. This work is motivated by infinite-dimensional applications, in particular to the Sard conjecture in sub-Riemannian geometry. To prove these results, we develop a general quantitative theory for Hausdorff approximations in semialgebraic geometry, which has independent interest.
Paper Structure (18 sections, 19 theorems, 118 equations)

This paper contains 18 sections, 19 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Hausdorff approximations
  3. Infinite--dimensional applications
  4. Acknowledgements
  5. Hausdorff approximations in semialgebraic geometry
  6. Semialgebraic sets and maps
  7. Some properties of semialgebraic sets
  8. Semialgebraic triviality
  9. Dimension and stratifications
  10. Thom-Milnor bound
  11. Hausdorff approximations using infinitesimals
  12. The evaluation map and the map ${\mathscr{L}}_0$
  13. Hausdorff limits
  14. Hausdorff approximations of closed and bounded sets
  15. Quantitative approximate definable choices
  16. ...and 3 more sections

Key Result

Theorem A

For every $c \in\mathbb{N}$ there exist $\kappa\in \mathbb{N}$ such that the following holds. Let $n,\ell,d\in \mathbb{N}$, with $1\leq \ell\leq n$. Let $\pi:\mathbb{R}^{n}\to \mathbb{R}^\ell$ be the projection onto the last $\ell$ coordinates and let $S\subset \mathbb{R}^{n}$ be a bounded closed se Then, for every $\epsilon>0$ there exists a closed semialgebraic set $A_\epsilon\subset \mathbb{R}^

Theorems & Definitions (61)

  • Theorem A: Quantitative approximate definable choice, first version
  • Theorem B: Quantitative approximate definable choice, second version
  • Remark 1: The case of polynomial maps
  • Theorem 2: LRT-Sard
  • Remark 3: On the use of the language of real closed fields
  • Definition 4: Algebraic Puiseux series
  • Remark 5: Algebraic Puiseux series as germs
  • Definition 6: Semialgebraic sets and maps
  • Definition 7: Diagram of a semialgebraic set
  • Definition 8: First--order formula
  • ...and 51 more