@article{BCL-Counting-Complexity-Classes-For-Numeric-Computations-Iii-Complex-Projective-Sets,
Title = {Counting Complexity Classes for Numeric Computations III: Complex Projective Sets},
Author = {Peter Bürgisser and Felipe Cucker and Martin Lotz},
Pages = {351-387},
Year = {2005},
Journal = {Foundations of Computational Mathematics},
Volume = {5},
Number = {4},
Abstract = {In [Bürgisser & Cucker 2004a] counting complexity classes $\#P_{\mathbb R}$ and $\#P_{\mathbb C}$ in the Blum-Shub-Smale setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [Bürgisser & Cucker 2004a] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class $FP_{\mathbb R}^{\#P_{\mathbb R}}$. In this paper, we prove that the corresponding result is true over $\mathbb C$, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class $FP_{\mathbb C}^{\#P_{\mathbb C}}$. We also obtain a corresponding completeness result for the Turing model.},
Url = {http://www3.math.tu-berlin.de/algebra/work/projective_rev3.pdf},
Url2 = {http://link.springer.com/article/10.1007%2Fs10208-005-0146-x}
}