量化近似可定义的选择
Antonio Lerario1, Luca Rizzi1, Daniele Tiberio1
1SISSA, via Bonomea 265, 34136 Trieste, Italy.
概括
研究人员在半代数几何学中开发了近似可定义的选择,提高了复杂性和变量依赖性. 这促进了无限维应用和子里曼几何学的发展.
科学领域:
- 半代数几何学的几何学
- 亚里曼的几何学子-里曼的几何学
- 真实代数几何学的几何学.
背景情况:
- 在半代数几何学中,投影是基本的.
- 可定义的选择在投射纤维内提供选择,但在计算上是复杂的.
- 现有的方法具有复杂性,随着变量数量的增加,其复杂性呈指数级.
研究的目的:
- 提高半代数几何学中可定义选择的复杂性.
- 开发具有更好的缩放性质的近似可定义选择.
- 为无限维度应用提供工具,包括Sard假设.
主要方法:
- 在半代数设置中开发一个对豪斯多夫近似的定量理论.
- 使用新技术构建近似选择.
- 对近似选择的复杂性和程度的分析.
主要成果:
- 构建了一个近似的可定义的选择.
- 这种近似选择的复杂性在投影的复杂性中是线性的.
- 复杂性独立于变量数量,这是一个显著的改进.
结论:
- 大致可定义的选择提供了一个更有效的替代方案,而不是精确的选择.
- 豪斯多夫近似的发展理论具有更广泛的含义.
- 这项工作促进了诸如亚里曼几何学和萨德推测等领域的进展.
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