Abstract
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.
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Beltran, C., Pardo, L. On Smale's 17th Problem: A Probabilistic Positive Solution. Found Comput Math 8, 1–43 (2008). https://doi.org/10.1007/s10208-005-0211-0
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DOI: https://doi.org/10.1007/s10208-005-0211-0