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

Authors:
Carlos Beltran;Luis Miguel Pardo
Affiliations:
Departamento de Matematicas, Estadistica y Computacion, F. de Ciencias, University of Cantabria, E-39071 Santander, Spain;Departamento de Matematicas, Estadistica y Computacion, F. de Ciencias, University of Cantabria, E-39071 Santander, Spain
Venue:
Foundations of Computational Mathematics
Year:
2008

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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.