On the Complexity of Solving Zero-Dimensional Polynomial Systems via Projection

TL;DR

This paper introduces a certified, complete method for solving zero-dimensional polynomial systems by projecting solutions into one dimension along multiple directions, enabling efficient reconstruction of all solutions.

Contribution

It presents a novel projection-based approach that is deterministic for solution projection and randomized for solution reconstruction, with explicit complexity bounds.

Findings

Method computes all complex solutions efficiently.

Provides bounds on bit complexity based on input parameters.

Uses a combination of deterministic and randomized steps.

Abstract

Given a zero-dimensional polynomial system consisting of n integer polynomials in n variables, we propose a certified and complete method to compute all complex solutions of the system as well as a corresponding separating linear form l with coefficients of small bit size. For computing l, we need to project the solutions into one dimension along O(n) distinct directions but no further algebraic manipulations. The solutions are then directly reconstructed from the considered projections. The first step is deterministic, whereas the second step uses randomization, thus being Las-Vegas. The theoretical analysis of our approach shows that the overall cost for the two problems considered above is dominated by the cost of carrying out the projections. We also give bounds on the bit complexity of our algorithms that are exclusively stated in terms of the number of variables, the total degree and the bitsize of the input polynomials.