Multipolynomial resultants and linear algebra

The problem of eliminating variables from a set of polynomial equations arises in many symbolic and numeric applications. The three main approaches are resultants, Grobner bases and the Wu-Ritt method. In practice, resultant based algorithms have been shown to be most effectice for certain applications. The result ant of a set of polynomial equations can be expressed in terms of matrices and determinants and the bottleneck in its comput ation is the symbolic expansion of determinants. In this paper we present interpolation based algorithms to compute symbolic determinants. The main characteristic of the algorithms is the use of techniques from linear algebra to reduce the symbolic complexity of the computation and number of function evaluations. These include linearizing a matrix polynomial to its companion form and similarity transformations for reduction to upper Hessenberg form followed by reduction to Frobenius canonical form. We consider dense as well as sparse interpolation algorithms. These algorithms have been implemented as part of a package for resultant computation and we discuss their performance for certain applications.

[1]  George Salmon.  Lessons introductory to the modern higher algebra , 1885 .

[2]  Xiao-Shan Gao,et al.  Implicitization of Rational Parametric Equations , 1992, J. Symb. Comput..

[3]  Dinesh Manocha,et al.  Efficient techniques for multipolynomial resultant algorithms , 1991, ISSAC '91.

[4]  A. L. Dixon.  The Eliminant of Three Quantics in two Independent Variables , 1909 .

[5]  Richard Zippel,et al.  Interpolating Polynomials from Their Values , 1990, J. Symb. Comput..

[6]  Dinesh Manocha,et al.  Multipolynomial Resultant Algorithms , 1993, J. Symb. Comput..

[7]  P. Lancaster,et al.  Factorization of selfadjoint matrix polynomials with constant signature , 1982 .

[8]  John F. Canny,et al.  Generalised Characteristic Polynomials , 1990, J. Symb. Comput..

[9]  Michael Ben-Or,et al.  A deterministic algorithm for sparse multivariate polynomial interpolation , 1988, STOC '88.

[10]  John F. Canny,et al.  Generalized Characteristic Polynomials , 1988, ISSAC.

[11]  F. S. Macaulay.  Some Formulæ in Elimination , 1902 .

[12]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[13]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[14]  Dinesh Manocha,et al.  Implicit Representation of Rational Parametric Surfaces , 1992, J. Symb. Comput..

[15]  Erich Kaltofen,et al.  Improved Sparse Multivariate Polynomial Interpolation Algorithms , 1988, ISSAC.

[16]  F. S. Macaulay,et al.  Note on the Resultant of a Number of Polynomials of the Same Degree , 1923 .

[17]  F. Morley,et al.  New Results in Elimination , 1927 .