Abstract |
We design a probabilistic algorithm that, given ϵ>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−ϵ. When applying this algorithm to u⋅F, where u is uniformly random in the product of unitary groups, the algorithm performs poly(n,δ)⋅L(F)⋅(Γ(F)logΓ(F)+log log ϵ^(−1)) operations on average. Here n is the number of variables, δ the maximum degree, L(F) denotes the evaluation cost of F, and Γ(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 poly(n,δ), the algorithm runs on average in time polynomial in n and δ. Our result may be interpreted as an affirmative answer to a refined version of Smale's 17th question, concerned with systems of \emphstructured polynomial equations. |