2011/01/17 by Angelos Mantzaflaris, Mantzaflaris, Angelos, Bernard Mourrain +1
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications #FOS: Computer and information sciences #Numerical Methods and Algorithms #Polynomial and algebraic computation #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.1101.3140
openalex publication_date 2011/01/17 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28
We develop a new symbolic-numeric algorithm for the certification of singular\nisolated points, using their associated local ring structure and certified\nnumerical computations. An improvement of an existing method to compute inverse\nsystems is presented, which avoids redundant computation and reduces the size\nof the intermediate linear systems to solve. We derive a one-step deflation\ntechnique, from the description of the multiplicity structure in terms of\ndifferentials. The deflated system can be used in Newton-based iterative\nschemes with quadratic convergence. Starting from a polynomial system and a\nsmall-enough neighborhood, we obtain a criterion for the existence and\nuniqueness of a singular root of a given multiplicity structure, applying a\nwell-chosen symbolic perturbation. Standard verification methods, based eg. on\ninterval arithmetic and a fixed point theorem, are employed to certify that\nthere exists a unique perturbed system with a singular root in the domain.\nApplications to topological degree computation and to the analysis of real\nbranches of an implicit curve illustrate the method.\n