Solving Systems of Polynomial Equations

  • Bhubaneswar Mishra
Part of the Texts and Monographs in Computer Science book series (MCS)

Abstract

The Gröbner basis algorithm can be seen to be a generalization of the classical Gaussian elimination algorithm from a set of linear multivariate polynomials to an arbitrary set of multivariate polynomials. The S-polynomial and reduction processes take the place of the pivoting step of the Gaussian algorithm. Taking this analogy much further, one can devise a constructive procedure to compute the set of solutions of a system of arbitrary multivariate polynomial equations:
$$ \begin{array}{*{20}{c}} {{f_1}\left( {{x_1}, \ldots ,{x_n}} \right) = 0,} \\ {{f_2}\left( {{x_1}, \ldots ,{x_n}} \right) = 0,} \\ \vdots \\ {{f_r}\left( {{x_1}, \ldots ,{x_n}} \right) = 0,} \end{array} $$
i.e., compute the set of points where all the polynomials vanish:
$$ \left\{\langle\xi_{1},\ldots,\xi_{n}\rangle:f_{i}(\xi_{1},\ldots,\xi_{n})=0,\quad{\rm for}\ {\rm all}\ 1\leq i\leq r\right\}. $$

Keywords

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Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Bhubaneswar Mishra
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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