| Abstract |
In [Bürgisser & Cucker 2004a] counting complexity classes $\#P_R$ and $\#P_C$ 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 $FP_R^R}$. 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 $FP_C^C}$. We also obtain a corresponding completeness result for the Turing model. |
