Approximate bivariate factorization: a geometric viewpoint

We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x; y]. Given a composite polynomial, stably square-free, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.

[1]  Lihong Zhi,et al.  Pseudofactors of multivariate polynomials , 2000, ISSAC.

[2]  Stephen M. Watt,et al.  The singular value decomposition for polynomial systems , 1995, ISSAC '95.

[3]  Zhonggang Zeng,et al.  The approximate GCD of inexact polynomials , 2004, ISSAC '04.

[4]  Takeo Ojika,et al.  Modified deflation algorithm for the solution of singular problems. I. A system of nonlinear algebraic equations , 1987 .

[5]  Marc Giusti,et al.  On Location and Approximation of Clusters of Zeros: Case of Embedding Dimension One , 2007, Found. Comput. Math..

[6]  Erich Kaltofen,et al.  On approximate irreducibility of polynomials in several variables , 2003, ISSAC '03.

[7]  A. Galligo.  Real Factorization of Multivariate Polynomials with Integer Coefficients , 2002 .

[8]  André Galligo,et al.  From an approximate to an exact absolute polynomial factorization , 2006, J. Symb. Comput..

[9]  Stephen M. Watt,et al.  A geometric-numeric algorithm for absolute factorization of multivariate polynomials , 2002, ISSAC '02.

[10]  Alexandru Dimca,et al.  Singularities and Topology of Hypersurfaces , 1992 .

[11]  Anton Leykin,et al.  Newton's method with deflation for isolated singularities of polynomial systems , 2006, Theor. Comput. Sci..

[12]  Erich Kaltofen,et al.  Efficient algorithms for computing the nearest polynomial with a real root and related problems , 1999, ISSAC '99.

[13]  David Rupprecht.  Elements de geometrie algebrique approchee : etude du pgcd et de la factorisation , 2000 .

[14]  I. Emiris,et al.  Certified approximate univariate GCDs , 1997 .

[15]  Tateaki Sasaki,et al.  Approximate factorization of multivariate polynomials and absolute irreducibility testing , 1991 .

[16]  Bernard Deconinck,et al.  Computing Riemann matrices of algebraic curves , 2001 .

[17]  Grégoire Lecerf,et al.  Lifting and recombination techniques for absolute factorization , 2007, J. Complex..

[18]  Erich Kaltofen,et al.  Polynomial Factorization 1987-1991 , 1992, LATIN.

[19]  Shuhong Gao,et al.  Factoring multivariate polynomials via partial differential equations , 2003, Math. Comput..

[20]  Tateaki Sasaki,et al.  A unified method for multivariate polynomial factorizations , 1993 .

[21]  George Labahn,et al.  Symbolic-numeric sparse interpolation of multivariate polynomials , 2006, ISSAC '06.

[22]  Erich Kaltofen,et al.  Approximate factorization of multivariate polynomials using singular value decomposition , 2008, J. Symb. Comput..

[23]  ChallengesPaul ZimmermannInria Lorrainezimmermann.  Polynomial Factorization , 1996 .

[24]  Stephen M. Watt,et al.  A numerical absolute primality test for bivariate polynomials , 1997, ISSAC.

[25]  Erich Kaltofen,et al.  Approximate factorization of multivariate polynomials via differential equations , 2004, ISSAC '04.

[26]  Erich Kaltofen,et al.  Challenges of Symbolic Computation: My Favorite Open Problems , 2000, J. Symb. Comput..

[27]  Andrew J. Sommese,et al.  Numerical factorization of multivariate complex polynomials , 2004, Theor. Comput. Sci..

[28]  Andrew J. Sommese,et al.  Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems , 2002, SIAM J. Numer. Anal..

[29]  Jan Verschelde,et al.  Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components , 2001 .

[30]  A. Galligo,et al.  Four lectures on polynomial absolute factorization , 2005 .

[31]  Stephen M. Watt,et al.  Towards factoring bivariate approximate polynomials , 2001, ISSAC '01.

[32]  André Galligo,et al.  Numerical Univariate Polynomial GCD , 1996 .

[33]  John F. Canny,et al.  Factoring Rational Polynomials Over the Complex Numbers , 1993, SIAM J. Comput..

[34]  Tateaki Sasaki,et al.  Analysis of approximate factorization algorithm I , 1992 .

[35]  Tateaki Sasaki,et al.  Approximate multivariate polynomial factorization based on zero-sum relations , 2001, ISSAC '01.