A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time

Pierre Lairez (1)

1 Institut für Mathematik [Berlin]

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.

Mots clés: Polynomial system; homotopy continuation; complexity; Smale's 17th problem; derandomization

Domaines: Analyse numérique [math.NA]; Complexité [cs.CC]

Pierre Lairez. A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time. Foundations of Computational Mathematics, 2016, ⟨10.1007/s10208-016-9319-7⟩. ⟨hal-01178588v3⟩