1997/02/08 by J. Maurice Rojas, Rojas, J. Maurice
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.math/9702222
openalex publication_date 1997/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over an algebraically closed field) of any n by n system of polynomial equations. Since we use the sparse resultant, we thus obtain complexity bounds (for converting any input polynomial system into a multilinear factorization problem) which are close to cubic in the degree of the underlying variety -- significantly better than previous bounds which were pseudo-polynomial in the classical Bézout bound. By carefully taking into account the underlying toric geometry, we are also able to improve the reliability of certain sparse resultant based algorithms for polynomial system solving.