2023/04/17 by Tianran Chen, Chen, Tianran
Computer Science · Engineering · Mathematics · #14Q15 #14Q65 #65H10 #65H14 #65H20 #Advanced Differential Equations and Dynamical Systems #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2304.08598
openalex publication_date 2023/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical methods. The polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets as numerical representations of non-isolated solution components, put forth by Sommese and Wampler, are two pillars of this field. In this paper, we show that a modified polyhedral homotopy places the computation of isolated solutions and sample sets for non-isolated components into a single, unified framework. In certain cases, this method also leads to a natural decomposition of the BKK bound into a sum of its local contributions.