Polynomial system solving has seen major progress in both theory and practice over the past decade. A landmark achievement was addressing Smale's 17th problem, establishing average-case polynomial-time algorithms for computing approximate solutions of polynomial systems via homotopy continuation. Recent improvements in complexity bounds for these algorithms led to the development of rigid homotopy methods. In this article, we prove a new complexity result for rigid homotopies for polynomial systems with Waring representations of prescribed length. In addition, we provide the first computational experiments for rigid homotopies using a preliminary implementation.

翻译：多项式系统求解在过去十年中在理论和实践方面均取得了重大进展。一项里程碑式的成就是解决了斯梅尔第17问题，通过同伦延拓建立了平均情形多项式时间算法，用于计算多项式系统的近似解。近期对这些算法复杂度界的改进催生了刚性同伦方法的发展。本文针对具有给定长度Waring表示的多项式系统，证明了刚性同伦的新复杂度结果。此外，我们利用初步实现，首次提供了刚性同伦的计算实验。