Solving via modular methods

TL;DR

This paper introduces a parallel modular algorithm that computes all solutions with multiplicities of zero-dimensional polynomial systems over rationals by combining modular methods with triangular decomposition and univariate solvers.

Contribution

It presents a novel parallel modular approach for solving polynomial systems, integrating M"oller's triangular decomposition with modular techniques.

Findings

Efficient computation of solutions with multiplicities.

Parallel modular algorithm improves performance.

Successfully applied to zero-dimensional polynomial systems.

Abstract

In this article we present a parallel modular algorithm to compute all solutions with multiplicities of a given zero-dimensional polynomial system of equations over the rationals. In fact, we compute a triangular decomposition using M\"oller's algorithm (cf. [M\"o93]) of the corresponding ideal in the polynomial ring over the rationals using modular methods, and then apply a solver for univariate polynomials.