Foundations of Computational Mathematics

, Volume 8, Issue 1, pp 1–43 | Cite as

On Smale's 17th Problem: A Probabilistic Positive Solution



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.


Positive Real Number Polynomial System Coarea Formula Approximate Zero Initial Pair 


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Copyright information

© Society for Foundations of Computational Mathematics 2007

Authors and Affiliations

  1. 1.Departamento de Matematicas, Estadistica y Computacion, F. de Ciencias, University of CantabriaE-39071 SantanderSpain

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