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