On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms
On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms
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关于对称可定义集的等变贝蒂数：消失、界限和算法
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
C. Riener
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文献类型：
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作者：
S. Basu;C. Riener
设$$mathrm {R}$$ R是一个真正的封闭场。证明了对于任意固定d，由以d为界的次多项式定义的$$mathrm {R}^k$$ Rk的闭对称半代数子集的等变有理上同群在d维及更大维上消失。这个消失的结果是紧密的。利用一种新的几何方法，我们还证明了$$d^{O(d)} s^d k^{lfloor d/2
floor -1} $$ dO(d)sdk在$$mathrm {R}^k$$ Rk的闭对称半代数子集的等变Betti数上的上界是⌊d/2⌋-1，这些子集是由无量子的公式定义的，这些公式涉及5个以d为界的度的对称多项式，其中$$1 < d ll s,k$$ 1<d≪s,k。这一界限紧密到仅取决于d的一个因素。这些结果显著改善了先前在Basu和Riener （Adv Math 305:803-855, 2017）中获得的结果，这些结果是使用不同的技术证明的。我们的新方法是相当普遍的，并且也给出了在$$mathrm {R}$$ r上任意o-极小结构中某些特殊类对称可定义集（在固定度的对称多项式映射下通过拉回对称的可定义集）的等变Betti数的界。最后，我们利用我们的新方法获得了计算这些等变Betti数的多项式有界复杂度的算法。相比之下，计算（不一定对称）半代数集的普通Betti数的问题被认为是一个棘手的问题，并且所有已知的算法都具有双指数复杂度。
Let $$mathrm {R}$$R be a real closed field. We prove that for any fixed d, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $$mathrm {R}^k$$Rk defined by polynomials of degrees bounded by d vanishes in dimensions d and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $$d^{O(d)} s^d k^{lfloor d/2
floor -1} $$dO(d)sdk⌊d/2⌋-1 on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $$mathrm {R}^k$$Rk defined by quantifier-free formulas involving s symmetric polynomials of degrees bounded by d, where $$1 < d ll s,k$$1<d≪s,k. This bound is tight up to a factor depending only on d. These results significantly improve upon those obtained previously in Basu and Riener (Adv Math 305:803–855, 2017) which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over $$mathrm {R}$$R. Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.