Counting complexity classes for numeric computations. III: Complex projective sets

In [8] counting complexity classes #P_ℝ and #P _ℂ in the Blum-Shub-Smale (BSS) setting of computations over the real and complex numbers, respectively, were introduced. One of the main results of [8] is that the problem to compute the Euler characteristic of a semialgebraic set is complete in the class FP_ℝ ^#Pℝ. In this paper, we prove that the corresponding result is true over ℂ, namely that the computation of the Euler characteristic of an affine or projective complex variety is complete in the class FP _ℂ^#Pℂ. We also obtain a corresponding completeness result for the Turing model. © 2005 SFoCM.