论文标题
有效的圆柱形细胞分解限制的限制次夫群
Effective cylindrical cell decompositions for restricted sub-Pfaffian sets
论文作者
论文摘要
由限制的PFAFFIAN功能（称为受限的亚帕夫式集合）产生的O最低结构，按照格式$ \ Mathcal {f} $的自然衡量复杂性，记录了涉及该集合定义的变量和量化器的数量，以及涉及该公式的$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $。 Khovanskii和后来的Gabrielov和Vorobjov已经建立了许多有效的估计值，以这些参数的范围来对亚福夫群的几何复杂性。在应用程序中，这些估计值在$ d $中是多项式，通常很重要。 尽管在该领域进行了大量研究，但仍不知道细胞分解，O-Wimimal几何形状的基础操作是否可以保留对$ d $的多项式依赖。我们稍微修改了通常的格式和程度概念，并证明了这些经过修订的概念实际上确实存在。结果是，我们还获得了第一个多项式（在$ d $中）的上限，以使用限制的亚菲达结构中使用量化公式定义的betti数量的总和。
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.
