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.

中文翻译：

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Hensel引理和Weierstrass准备工作的复杂性和并行执行

Hensel引理与Weierstrass准备定理的重复应用相结合，可以对具有多元幂级数系数的多项式进行因式分解。我们提供了此方法的复杂性分析，并利用这些结果来指导并行实现的负载平衡以同时更新所有因素。尤其是，因式分解会创建一条流水线，其中同时计算第一因子的度数k和第二因子的度数k-1等。实现挑战是因子之间的计算工作固有的不规则性，例如我们的复杂性分析揭示了这一点。通过Weierstrass准备工作的并行化，可以实现额外的资源利用和负载平衡。实验结果证明了这种混合并行方案的有效性，