Skip to main content

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

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].

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding authors

Correspondence to Peter Burgisser or Martin Lotz.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Burgisser, P., Lotz, M. The Complexity of Computing the Hilbert Polynomial of Smooth Equidimensional Complex Projective Varieties. Found Comput Math 7, 59–86 (2007). https://doi.org/10.1007/s10208-005-0175-0

Download citation

Keywords

  • Chern Class
  • Schubert Variety
  • Homogeneous Ideal
  • Elementary Symmetric Function
  • Hilbert Polynomial