Quantitative approximate definable choices

TL;DR

This paper introduces an approximate definable choice in semialgebraic geometry that reduces complexity from exponential to linear in the projection's degree, with applications to infinite-dimensional problems.

Contribution

It develops a quantitative theory for Hausdorff approximations, enabling approximate definable choices with complexity independent of variable count.

Findings

Constructed approximate selections with linear degree complexity

Reduced complexity dependence from exponential to linear

Applied results to the Sard conjecture in sub-Riemannian geometry

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.