多项式系统解集的分解：一种新的并行单演分裂算法
Decomposing Solution Sets of Polynomial Systems: A New Parallel Monodromy Breakup Algorithm
AI总结：
本文提出一种新的并行单演分裂算法，用于多项式系统正维解集的数值不可约分解，通过避免不必要的同伦路径和绕过同步问题，在较大次的解集上性能更优。
摘要：
我们考虑将多项式系统的正维解集数值不可约分解为不可约因子。路径跟踪技术计算围绕奇点的环，将同一不可约分量上的点连接起来。对每个因子计算线性迹可验证分解的正确性。该分解方法在相对较高次的解集上表现出良好的实际性能。\n利用单演和线性迹的相同概念，我们提出了一种新的单演分裂算法。该算法比旧方法性能更优，旧方法需要构造见证点的排列来分解解集。相比之下，新算法采用更精细的方法，使我们能够避免跟踪不必要的同伦路径。\n由于我们在设计串行算法时就考虑了分布式计算，因此其并行版本易于构建。同步问题导致了旧算法直接并行版本的性能损失。我们新方法的并行实现绕过了这些问题，因此表现出更好的性能，尤其是在较大次的解集上。
英文摘要：
We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible components. The computation of a linear trace for each factor certifies the decomposition. This factorization method exhibits a good practical performance on solution sets of relative high degrees. Using the same concepts of monodromy and linear trace, we present a new monodromy breakup algorithm. It shows a better performance than the old method which requires construction of permutations of witness points in order to break up the solution set. In contrast, the new algorithm assumes a finer approach allowing us to avoid tracking unnecessary homotopy paths. As we designed the serial algorithm keeping in mind distributed computing, an additional advantage is that its parallel version can be easily built. Synchronization issues resulted in a performance loss of the straightforward parallel version of the old algorithm. Our parallel implementation of the new approach bypasses these issues, therefore, exhibiting a better performance, especially on solution sets of larger degree.
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