Geometric completion of differential systems using numeric-symbolic continuation
ACM SIGSAM Bulletin
Irreducible decomposition of curves
Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
Verifying topological indices for higher-order rank deficiencies
Journal of Complexity
A geometric-numeric algorithm for absolute factorization of multivariate polynomials
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
A complete symbolic-numeric linear method for camera pose determination
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Computer algebra handbook
Deformation techniques to solve generalised Pham systems
Theoretical Computer Science - Algebraic and numerical algorithm
Numerical factorization of multivariate complex polynomials
Theoretical Computer Science - Algebraic and numerical algorithm
Computing the multiplicity structure in solving polynomial systems
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
On approximate triangular decompositions in dimension zero
Journal of Symbolic Computation
Numerical algebraic geometry and kinematics
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Component-level parallelization of triangular decompositions
Proceedings of the 2007 international workshop on Parallel symbolic computation
Heuristics to accelerate the Dixon resultant
Mathematics and Computers in Simulation
Numerical primary decomposition
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library
ICCS '07 Proceedings of the 7th international conference on Computational Science, Part II
Symbolic-numeric sparse interpolation of multivariate polynomials
Journal of Symbolic Computation
Towards toric absolute factorization
Journal of Symbolic Computation
Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm
International Journal of Computational Science and Engineering
Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Comparing acceleration techniques for the Dixon and Macaulay resultants
Mathematics and Computers in Simulation
Chern numbers of smooth varieties via homotopy continuation and intersection theory
Journal of Symbolic Computation
Diversification improves interpolation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Sampling algebraic sets in local intrinsic coordinates
Computers & Mathematics with Applications
Parallel homotopy algorithms to solve polynomial systems
ICMS'06 Proceedings of the Second international conference on Mathematical Software
Interfacing with the numerical homotopy algorithms in PHCpack
ICMS'06 Proceedings of the Second international conference on Mathematical Software
Implicit reduced involutive forms and their application to engineering multibody systems
IWMM'04/GIAE'04 Proceedings of the 6th international conference on Computer Algebra and Geometric Algebra with Applications
Hybrid method for solving new pose estimation equation system
IWMM'04/GIAE'04 Proceedings of the 6th international conference on Computer Algebra and Geometric Algebra with Applications
Numerical calculation of H-bases for positive dimensional varieties
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Computing Puiseux series for algebraic surfaces
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
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
Hi-index | 0.00 |
In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials.