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Foundations of Computational Mathematics

, Volume 7, Issue 1, pp 59–86 | Cite as

The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties

  • Peter BurgisserEmail author
  • Martin LotzEmail author
Article

Abstract

We continue the study of counting complexity begun in [13], [14], [15] 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. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. 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 in Bach [1].

Keywords

Chern Class Schubert Variety Homogeneous Ideal Elementary Symmetric Function Hilbert Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Society for the Foundations of Computational Mathematics 2006

Authors and Affiliations

  1. 1.Institute of Mathematics, University of Paderborn, D-33095PaderbornGermany
  2. 2.Department of Mathematics, City University of Hong Kong, 83 Tat Chee AvenueKowloonHong Kong

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