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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

Polynomial Ring Hamiltonian Path Residue Class Truth Assignment Common Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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