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 exploited 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.

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的多项式方程组求解方法。胡贝尔与施图姆费尔斯提出的多面体同伦算法将该领域与代数几何的现代发展紧密联系起来——该算法通过利用方程的组合结构，催生了高效的多项式方程求解软件。此后，数值非线性代数的发展始终受代数几何及其应用领域的滋养，衍生出新的求解方法、正维数簇研究算法、结果验证技术，以及数学内外的广泛实际应用。随着新型实现工具的出现，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。