Condition and homology in semialgebraic geometry
Condition and homology in semialgebraic geometry
复制标题
半代数几何中的条件和同调
DOI：
10.14279/depositonce-9453
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Josué Tonelli
中科院分区：
文献类型：
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作者：
Josué Tonelli
半分布集的同源组（由布尔公式给出）的计算仍然是计算半几何学的开放挑战之一。尽管仅在变量数量上寻找一种单一的指数时间，但截至目前，现有算法是象征性的，并且指数倍增。在本博士学位论文中，我们展示了如何获得在单个指数时间内运行的数值算法，并具有很高的概率，从而改善了最新时间。为此，我们解释了来自数值代数几何形状，数值复杂性和拓扑数据分析的基本思想，方法和技术，从而使这一进展成为可能。我们结束了一系列的开放问题和问题，指出了拓扑不变的数值计算的可能未来。此外，在附录中，我们介绍了随机次务系统的预期零零数的主题，并为西班牙语中的中心主题提供了可访问的帐户。
The computation of the homology groups of semialgebraic sets (given by Boolean formulas) remains one of the open challenges of computational semialgebraic geometry. Despite the search for an algorithm taking singly exponential time only on the number of variables, as of today, the existing algorithms are symbolic and doubly exponential. In this PhD thesis, we show how to obtain a numerical algorithm running in single exponential time with very high probability, which improves the state-of-the-art. To do so, we explain the underlying ideas, methods and techniques from numerical algebraic geometry, numerical complexity and topological data analysis that made this progress possible. We finish with a list of open problems and questions pointing to a possible future of the numerical computation of topological invariants. Additionally, in the appendices, we cover the topic of the expected number of real zeros of a random fewnomial system and we give an accessible account of the central theme in Spanish.