On the Complexity and Parallel Implementation of Hensel's Lemma and Weierstrass Preparation

TL;DR

This paper analyzes the complexity of factorizing multivariate polynomials using Hensel's lemma and Weierstrass preparation, and introduces a parallel implementation that significantly improves computational efficiency.

Contribution

It provides a detailed complexity analysis and a novel parallel implementation for polynomial factorization using Hensel's lemma and Weierstrass preparation.

Findings

Achieves up to 9x speedup on 12 cores

Demonstrates effective load-balancing in parallelization

Validates the approach with experimental results

Abstract

Hensel's lemma, combined with repeated applications of Weierstrass preparation theorem, allows for the factorization of polynomials with multivariate power series coefficients. We present a complexity analysis for this method and leverage those results to guide the load-balancing of a parallel implementation to concurrently update all factors. In particular, the factorization creates a pipeline where the terms of degree k of the first factor are computed simultaneously with the terms of degree k-1 of the second factor, etc. An implementation challenge is the inherent irregularity of computational work between factors, as our complexity analysis reveals. Additional resource utilization and load-balancing is achieved through the parallelization of Weierstrass preparation. Experimental results show the efficacy of this mixed parallel scheme, achieving up to 9x parallel speedup on 12 cores.