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Quantitative approximate definable choices.

Antonio Lerario1, Luca Rizzi1, Daniele Tiberio1

  • 1SISSA, via Bonomea 265, 34136 Trieste, Italy.

Mathematische Annalen
|April 7, 2025
PubMed
Summary
This summary is machine-generated.

Researchers developed approximate definable choices in semialgebraic geometry, improving complexity and variable dependence. This advances infinite-dimensional applications and sub-Riemannian geometry.

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Area of Science:

  • Semialgebraic geometry
  • Sub-Riemannian geometry
  • Real algebraic geometry

Background:

  • Projections are fundamental in semialgebraic geometry.
  • Definable choices provide selections within projection fibers but are computationally complex.
  • Existing methods have complexity that scales exponentially with the number of variables.

Purpose of the Study:

  • To improve the complexity of definable choices in semialgebraic geometry.
  • To develop approximate definable choices with better scaling properties.
  • To provide tools for infinite-dimensional applications, including the Sard conjecture.

Main Methods:

  • Development of a quantitative theory for Hausdorff approximations in semialgebraic settings.
  • Construction of approximate selections using novel techniques.
  • Analysis of the complexity and degree of the approximate selections.

Main Results:

  • An approximate definable choice is constructed.
  • The complexity of this approximate selection is linear in the projection's complexity.
  • The complexity is independent of the number of variables, a significant improvement.

Conclusions:

  • Approximate definable choices offer a more efficient alternative to exact choices.
  • The developed theory of Hausdorff approximations has broader implications.
  • This work facilitates progress in areas like sub-Riemannian geometry and the Sard conjecture.