Numerical factorization of multivariate complex polynomials

Authors:
Andrew J. Sommese;Jan Verschelde;Charles W. Wampler
Affiliations:
Department of Mathematics, University of Notre Dame, Notre Dame, IN;Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan (M/C 249), Chicago, IL;General Motors Research & Development, Mail Code 480-106-359, 30500 Mound Road, Warren, MI
Venue:
Theoretical Computer Science - Algebraic and numerical algorithm
Year:
2004

Citing 17
Cited 15

Factoring rational polynomials over the complex numbers

SIAM Journal on Computing
Solving a Polynomial Equation: Some History and Recent Progress

SIAM Review
Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation

ACM Transactions on Mathematical Software (TOMS)
Challenges of symbolic computation: my favorite open problems

Journal of Symbolic Computation
Pseudofactors of multivariate polynomials

ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Numerical homotopies to compute generic points on positive dimensional algebraic sets

Journal of Complexity
Finding a cluster of zeros of univariate polynomials

Journal of Complexity
Computation of approximate polynomial GCDs and an extension

Information and Computation
Towards factoring bivariate approximate polynomials

Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Semi-numerical determination of irreducible branches of a reduced space curve

Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Approximate multivariate polynomial factorization based on zero-sum relations

Proceedings of the 2001 international symposium on Symbolic and algebraic computation
Accuracy and Stability of Numerical Algorithms

Accuracy and Stability of Numerical Algorithms
Parallel Robots

Parallel Robots
Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

SIAM Journal on Numerical Analysis
Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems

SIAM Journal on Numerical Analysis
Irreducible decomposition of curves

Journal of Symbolic Computation - Computer algebra: Selected papers from ISSAC 2001
A geometric-numeric algorithm for absolute factorization of multivariate polynomials

Proceedings of the 2002 international symposium on Symbolic and algebraic computation

Approximate factorization of multivariate polynomials via differential equations

ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Symbolic-numeric sparse interpolation of multivariate polynomials

Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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
Computing monodromy groups defined by plane algebraic curves

Proceedings of the 2007 international workshop on Symbolic-numeric computation
Approximate factorization of multivariate polynomials using singular value decomposition

Journal of Symbolic Computation
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
Extracting numerical factors of multivariate polynomials from taylor expansions

Proceedings of the 2009 conference on Symbolic numeric computation
From an approximate to an exact absolute polynomial factorization

Journal of Symbolic Computation
Modular Las Vegas algorithms for polynomial absolute factorization

Journal of Symbolic Computation
Reliable root detection with the qd-algorithm: When Bernoulli, Hadamard and Rutishauser cooperate

Applied Numerical Mathematics
Tropical algebraic geometry in Maple: A preprocessing algorithm for finding common factors for multivariate polynomials with approximate coefficients

Journal of Symbolic Computation
Diversification improves interpolation

Proceedings of the 36th international symposium on Symbolic and algebraic computation

Quantified Score

Hi-index	0.00

Visualization

Abstract

One can consider the problem of factoring multivariate complex polynomials as a special case of the decomposition of a pure dimensional solution set of a polynomial system into irreducible components. The importance and nature of this problem however justify a special treatment. We exploit the reduction to the univariate root finding problem as a way to sample the polynomial more efficiently, certify the decomposition with linear traces, and apply interpolation techniques to construct the irreducible factors. With a random combination of differentials we lower multiplicities and reduce to the regular case. Estimates on the location of the zeroes of the derivative of polynomials provide bounds on the required precision. We apply our software to study the singularities of Stewart-Gough platforms.