Foundations of Computational Mathematics

, Volume 17, Issue 5, pp 1265–1292 | Cite as

A Deterministic Algorithm to Compute Approximate Roots of Polynomial Systems in Polynomial Average Time



We describe a deterministic algorithm that computes an approximate root of n complex polynomial equations in n unknowns in average polynomial time with respect to the size of the input, in the Blum–Shub–Smale model with square root. It rests upon a derandomization of an algorithm of Beltrán and Pardo and gives a deterministic affirmative answer to Smale’s 17th problem. The main idea is to make use of the randomness contained in the input itself.


Polynomial system Homotopy continuation Complexity  Smale’s 17th problem Derandomization 

Mathematics Subject Classification

Primary 68Q25 Secondary 65H10 65H20 65Y20 



I am very grateful to Peter Bürgisser for his help and constant support and to Carlos Beltrán for having carefully commented this work. I thank the two referees for their meticulous reading and their insightful suggestions.


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Copyright information

© SFoCM 2016

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany

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