Smale’s 17th problem: Average polynomial time to compute affine and projective solutions

Beltrán, Carlos; Pardo, Luis

doi:10.1090/S0894-0347-08-00630-9

Smale’s 17th problem: Average polynomial time to compute affine and projective solutions
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by Carlos Beltrán and Luis Miguel Pardo;

J. Amer. Math. Soc. 22 (2009), 363-385

DOI: https://doi.org/10.1090/S0894-0347-08-00630-9

Published electronically: November 6, 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, in polynomial time with a uniform algorithm?” We give a positive answer to this question. Namely, we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations $f:\mathbb {C}^n\longrightarrow \mathbb {C}^n$, performing a number of arithmetic operations which is polynomial in the size of the input, on the average.

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