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 speedup on 12 cores.

翻译：Hensel的 Lemma,加上Weierstrass准备理论的反复应用,使得多分子制和多变动力序列系数的因子化成为可能。我们对这一方法进行了复杂分析,并利用这些结果指导平行执行的负平衡,同时更新所有因素。特别是,这个因子的因子因子因子因子因子因子因子因子因子因子因子因子因子K-1而计算而同时计算k级条件的管道。正如我们的复杂性分析所揭示的那样,执行方面的挑战是各种因素之间计算工作固有的不规则性。额外的资源利用和负载平衡是通过Weierstras制的并行制备来实现的。实验结果显示这一混合平行计划的效果,在12个核心上达到9x加速。