## The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties (2005)

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Venue: | Foundations of Computational Mathematics |

Citations: | 3 - 2 self |

### BibTeX

@TECHREPORT{Bürgisser05thecomplexity,

author = {Peter Bürgisser and Martin Lotz},

title = {The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties},

institution = {Foundations of Computational Mathematics},

year = {2005}

}

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### Abstract

We continue the study of counting complexity begun in [11, 14, 13] 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 in Bach [1]. 1