光滑等维复射影簇的Hilbert多项式的计算复杂性
The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties
摘要：
我们继续研究计数的复杂性开始在[Buergisser，Cucker 04]和[Buergisser，Cucker，Lotz 05]通过证明上界和下界的计算希尔伯特多项式的齐次理想的复杂性。我们证明了计算光滑等维复射影簇的Hilbert多项式的问题可以在多项式时间内归结为计算多元多项式的有限集合的复公共零点的个数的问题。此外，我们证明了计算齐次理想的Hilbert多项式的更一般的问题是多项式空间困难的。这意味着计算射影空间上相干层的上同调群的秩和欧拉特征的问题的多项式空间下界，改进了Bach（JSC 1999）的P-下界。
英文摘要：
We continue the study of counting complexity begun in [Buergisser, Cucker 04] and [Buergisser, Cucker, Lotz 05] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound of Bach (JSC 1999).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
