Title: Quantitative approximate definable choices Show Chinese title

Title: 定量近似可定义选择

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. Show Chinese abstract

Abstract: 在半代数几何中，投影起着重要作用。 一个可定义的选择是投影每个纤维中一点的半代数选择。 根据半代数平凡性，可定义的选择存在，但其复杂度随变量数量呈指数增长。 通过允许选择是近似的（在豪斯多夫意义下），我们改进了这一结果。 特别是，我们构造了一个近似选择，其次数与投影的复杂度成线性关系，并且不依赖于变量的数量。 这项工作受到无限维应用的启发，特别是对子黎曼几何中的Sard猜想。 为了证明这些结果，我们开发了一种关于半代数几何中豪斯多夫逼近的一般定量理论，这具有独立的兴趣。