定量近似可定义的选择。

IF 1.3 2区 数学 Q1 MATHEMATICS
Mathematische Annalen Pub Date : 2025-01-01 Epub Date: 2025-03-09 DOI:10.1007/s00208-025-03128-3
Antonio Lerario, Luca Rizzi, Daniele Tiberio
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引用次数: 0

摘要

在半代数几何中,投影起着重要的作用。可定义的选择是在投影的每个纤维中选取一个点的半代数选择。可定义的选择存在于半代数的琐碎性中,但其复杂性依赖于变量的数量。通过允许选择是近似的(在Hausdorff意义上),我们改进了这个结果。特别地,我们构造了一个近似选择,它的程度在投影的复杂性上是线性的,并且不依赖于变量的数量。这项工作的动机是无限维的应用,特别是在亚黎曼几何中的Sard猜想。为了证明这些结果,我们发展了半代数几何中具有独立意义的Hausdorff近似的一般定量理论。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Quantitative approximate definable choices.

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.

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来源期刊
Mathematische Annalen
Mathematische Annalen 数学-数学
CiteScore
2.90
自引率
7.10%
发文量
181
审稿时长
4-8 weeks
期刊介绍: Begründet 1868 durch Alfred Clebsch und Carl Neumann. Fortgeführt durch Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguignon, Wolfgang Lück und Nigel Hitchin. The journal Mathematische Annalen was founded in 1868 by Alfred Clebsch and Carl Neumann. It was continued by Felix Klein, David Hilbert, Otto Blumenthal, Erich Hecke, Heinrich Behnke, Hans Grauert, Heinz Bauer, Herbert Amann, Jean-Pierre Bourguigon, Wolfgang Lück and Nigel Hitchin. Since 1868 the name Mathematische Annalen stands for a long tradition and high quality in the publication of mathematical research articles. Mathematische Annalen is designed not as a specialized journal but covers a wide spectrum of modern mathematics.
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