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 17th Problem.

• Type: Journal
• Year: 2008
• Where: FOCM
• Authors: Carlos Beltrán, Luis Miguel Pardo
• Keywords: 17th Problem, Complex Polynomial Equations, FOCM 2008, Polynomial