Geometric completion of differential systems using numeric-symbolic continuation
ACM SIGSAM Bulletin
Absolute polynomial factorization in two variables and the knapsack problem
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Numerical factorization of multivariate complex polynomials
Theoretical Computer Science - Algebraic and numerical algorithm
An intrinsic homotopy for intersecting algebraic varieties
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Lifting and recombination techniques for absolute factorization
Journal of Complexity
Approximate bivariate factorization: a geometric viewpoint
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Numerical algebraic geometry and kinematics
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm
International Journal of Computational Science and Engineering
Continuations and monodromy on random riemann surfaces
Proceedings of the 2009 conference on Symbolic numeric computation
An intrinsic homotopy for intersecting algebraic varieties
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Computing monodromy via continuation methods on random Riemann surfaces
Theoretical Computer Science
Parallel homotopy algorithms to solve polynomial systems
ICMS'06 Proceedings of the Second international conference on Mathematical Software
Numerically Computing Real Points on Algebraic Sets
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Cell decomposition of almost smooth real algebraic surfaces
Numerical Algorithms
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Many polynomial systems have solution sets comprised of multiple irreducible components, possibly of different dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using floating-point numerical processes, into its components. Prior work has shown how to generate sets of generic points guaranteed to include points from every component. Furthermore, we have shown how monodromy can be used to efficiently predict the partition of these points by membership in the components. However, confirmation of this prediction required an expensive procedure of sampling each component to find an interpolating polynomial that vanishes on it. This paper proves theoretically and demonstrates in practice that linear traces suffice for this verification step, which gives great improvement in both computational speed and numerical stability. Moreover, in the case that one may still wish to compute an interpolating polynomial, we show how to do so more efficiently by building a structured grid of samples, using divided differences, and applying symmetric functions. Several test problems illustrate the effectiveness of the new methods.