A softly optimal Monte Carlo algorithm for solving bivariate polynomial systems over the integers

Abstract

We give an algorithm for the symbolic solution of polynomial systems in Z[X,Y]. Following previous work with Lebreton, we use a combination of lifting and modular composition techniques, relying in particular on Kedlaya and Umans’ recent quasi-linear time modular composition algorithm.The main contribution in this paper is an adaptation of a deflation algorithm of Lecerf, that allows us to treat singular solutions for essentially the same cost as the regular ones. Altogether, for an input system with degree d and coefficients of bit-size h, we obtain Monte Carlo algorithms that achieve probability of success at least 1−1/2P, with running time d2+εÕ(d2+dh+dP+P2) bit operations, for any ε>0, where the Õ notation indicates that we omit polylogarithmic factors.