Gröbner duality and multiplicities in polynomial system solving
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Analysis of zero clusters in multivariate polynomial systems
ISSAC '96 Proceedings of the 1996 international symposium on Symbolic and algebraic computation
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Newton's method with deflation for isolated singularities of polynomial systems
Theoretical Computer Science
On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One
Foundations of Computational Mathematics
Computing the multiplicity structure from geometric involutive form
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Topology and arrangement computation of semi-algebraic planar curves
Computer Aided Geometric Design
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
Nearest multivariate system with given root multiplicities
Journal of Symbolic Computation
Continued fraction expansion of real roots of polynomial systems
Proceedings of the 2009 conference on Symbolic numeric computation
Approximating algebraic space curves by circular arcs
Proceedings of the 7th international conference on Curves and Surfaces
An improved method for evaluating Max Noether conditions: case of breadth one
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Approximately singular systems and ill-conditioned polynomial systems
CASC'12 Proceedings of the 14th international conference on Computer Algebra in Scientific Computing
Verified error bounds for isolated singular solutions of polynomial systems: Case of breadth one
Theoretical Computer Science
Verified error bounds for real solutions of positive-dimensional polynomial systems
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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We develop a new symbolic-numeric algorithm for the certification of singular isolated points, using their associated local ring structure and certified numerical computations. An improvement of an existing method to compute inverse systems is presented, which avoids redundant computation and reduces the size of the intermediate linear systems to solve. We derive a one-step deflation technique, from the description of the multiplicity structure in terms of differentials. The deflated system can be used in Newton-based iterative schemes with quadratic convergence. Starting from a polynomial system and a sufficiently small neighborhood, we obtain a criterion for the existence and uniqueness of a singular root of a given multiplicity structure, applying a well-chosen symbolic perturbation. Standard verification methods, based e.g. on interval arithmetic and a fixed point theorem, are employed to certify that there exists a unique perturbed system with a singular root in the domain. Applications to topological degree computation and to the analysis of real branches of an implicit curve illustrate the method.