On Smale's 17th Problem: A Probabilistic Positive Solution
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- Beltran, C. & Pardo, L. Found Comput Math (2008) 8: 1. doi:10.1007/s10208-005-0211-0
Smale's 17th Problem asks “Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm?” We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a partial positive solution to Smale's 17th Problem.