限制子Pff集的有效圆柱胞腔剖分
Effective cylindrical cell decompositions for restricted sub-Pfaffian sets
摘要：
由限制性Pfweian函数生成的o-最小结构，称为限制性子Pfweian集合，在格式$\mathcal{F}$方面承认复杂性的自然度量，记录信息，如在集合的定义中涉及的变量和量词的数量，以及记录所涉及的方程的度的度$D$。Khovanskii和后来的Gabrielov和Vorobjov已经建立了许多有效的估计几何复杂性的sub-Pfrons集在这些参数。在应用中，这些估计在$D$中是多项式的通常是重要的。尽管在这方面做了大量的研究，它仍然是不知道是否胞腔剖分，O-最小几何的基本操作，保持多项式依赖于$D$。我们稍微修改了通常的格式和程度的概念，并证明，与这些修订后的概念，这实际上举行。作为一个结果，我们还获得了第一个多项式（在$D$）上限的总和贝蒂数的限制子Pfiran结构中使用量化公式定义的集合。
英文摘要：
The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format $\mathcal{F}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree $D$ recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions this does in fact hold. As one consequence we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.
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