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%DIFDELCMD < \left( %%% \Gamma(F) \log \Gamma(F) + \log \log \epsilon^{-1 }%DIFDELCMD < \right) %%% operations on average. Here n is the number of variables, \delta the maximum degree, L(F) denotes the evaluation cost of F, and \Gamma(F) reflects an aspect of the numerical condition of F. Moreover, we prove that for inputs given by random Gaussian algebraic branching programs of size (n,\delta), the algorithm runs on average in time polynomial in n and \delta. Our result may be interpreted as an affirmative answer to a refined version of  Smale's 17th question, concerned with systems of\emph{structured polynomial equations .
This work studies the average complexity of solving structured polynomial systems that are characterized by a low evaluation cost, as opposed to the dense random model previously used. Firstly, we design a continuation algorithm that computes, with high probability, an approximate zero of a polynomial system given only as black-box evaluation program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute an approximate zero of a random structured polynomial system with n equations of degree at most \delta} in n variables with only poly (n, %DIFDELCMD < \left( %%% }%DIFDELCMD < \right) %%% \delta ) L operations with high probability. This exceeds the expectations implicit in Smale's 17th \emph{ problem .
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program. Secondly, we introduce a universal model of random polynomial systems with prescribed evaluation complexity L. Combining both, we show that we can compute
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