A Blackbox Polynomial System Solver on Parallel Shared Memory Computers

TL;DR

This paper presents a parallel shared memory implementation of a blackbox polynomial system solver using homotopy continuation, enabling efficient numerical irreducible decomposition of polynomial systems on multicore computers.

Contribution

It introduces a novel parallel algorithm for polynomial system solving that improves load balancing and pipelining in shared memory environments.

Findings

Successfully applied to cyclic n-roots problems for n=8, 9, 12

Achieved efficient load balancing and pipelining in parallel implementation

Demonstrated effectiveness on multicore shared memory systems

Abstract

A numerical irreducible decomposition for a polynomial system provides representations for the irreducible factors of all positive dimensional solution sets of the system, separated from its isolated solutions. Homotopy continuation methods are applied to compute a numerical irreducible decomposition. Load balancing and pipelining are techniques in a parallel implementation on a computer with multicore processors. The application of the parallel algorithms is illustrated on solving the cyclic $n$-roots problems, in particular for $n = 8, 9$, and~12.