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

• Published in: Journal of Complexity Vol. 34; pp. 78 - 128
• Main Authors: Mehrabi, Esmaeil, Schost, Éric
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-06-2016
• Subjects: Algorithm; Bivariate system; Complexity; Bivariate system; Algorithm; Complexity
• ISSN: 0885-064X; 1090-2708
• DOI: 10.1016/j.jco.2015.11.009

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.