Before revision: We design a probabilistic algorithm that , given \epsilon>0 and a polynomial system F given by black-box evaluation functions, outputs an approximate zero of F, in the sense of Smale, with probability at least 1-\epsilon. When applying this algorithm to u \cdot F, where u is uniformly random in the product of unitary groups, the algorithm performs }  (n, \delta) \cdot L (F) \cdot \Gamma(F) \log \Gamma(F) + \log \log \epsilon^{-1 } operations on average.
After revision: We design a probabilistic algorithm that , given \epsilon>0 and a polynomial system F given by black-box evaluation functions, outputs an approximate zero of a random structured polynomial system with n equations of degree at most } (n, \delta) \cdot L (F) \cdot \Gamma(F) \log \Gamma(F) + \log \log \epsilon^{-1 } operations on average.