Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of B\'ezout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.

翻译：数值非线性代数是一种利用数值分析研究多项式方程的计算范式。其起源可追溯至基于经典贝祖定理求解多项式方程组的方法。胡贝尔与斯图姆费尔斯提出的多面体同伦算法将这一领域与代数几何的现代发展紧密联系起来——该算法利用了方程的组合结构，并促成了求解多项式方程的高效软件。此后，数值非线性代数的持续发展始终以代数几何及其应用为导向，衍生出包括新型求解方法、正维数簇研究算法、结果验证技术，以及数学内外领域的广泛应用。随着新实现方案的问世，数值非线性代数现已成为代数几何及其应用领域的基础计算工具。本文综述了部分创新成果及近期应用案例。