On the bit complexity of polynomial system solving

TL;DR

This paper introduces a probabilistic algorithm for solving polynomial systems over the rationals with quadratic bit complexity relative to the system's Bézout number, utilizing prime reduction and p-adic lifting techniques.

Contribution

The paper presents a novel probabilistic algorithm with quadratic bit complexity for solving polynomial systems, including new results on prime selection and geometric property preservation.

Findings

Algorithm has quadratic complexity in Bézout number

Establishes bounds on the bit length of 'lucky' primes

Uses Chow forms and Nullstellensatz for analysis

Abstract

We exhibit a probabilistic algorithm which solves a polynomial system over the rationals defined by a reduced regular sequence. Its bit complexity is roughly quadratic in the B\'ezout number of the system and linear in its bit size. Our algorithm solves the input system modulo a prime number p and applies p-adic lifting. For this purpose, we establish a number of results on the bit length of a "lucky" prime p, namely one for which the reduction of the input system modulo p preserves certain fundamental geometric and algebraic properties of the original system. These results rely on the analysis of Chow forms associated to the set of solutions of the input system and effective arithmetic Nullstellens\"atze.