A deterministic algorithm to compute approximate roots of polynomial systems in polynomial average time
It resolves a long-standing open problem in computational complexity for polynomial systems, offering a deterministic solution where only probabilistic ones existed.
The paper presents a deterministic algorithm that computes approximate roots of polynomial systems in average polynomial time, providing a deterministic affirmative answer to Smale's 17th problem.
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.