A Gröbner basis for a system of Polynomial equations is an equivalence system that possesses useful properties. It is very roughly analogous to computing an Orthonormal Basis from a set of Basis Vectors and can be described roughly as a combination of Gaussian Elimination (for linear systems) and the Euclidean Algorithm (for Univariate Polynomials over a Field).

Gröbner bases are useful in the construction of symbolic algebra algorithms. The algorithm for computing Gröbner bases is known as Buchberger's Algorithm.

**References**

Adams, W. W. and Loustaunau, P. *An Introduction to Gröbner Bases.*
Providence, RI: Amer. Math. Soc., 1994.

Becker, T. and Weispfennig, V. *Gröbner Bases: A Computational Approach to Commutative Algebra*. New York: Springer-Verlag, 1993.

Cox, D.; Little, J.; and O'Shea, D. *Ideals, Varieties, and Algorithms: An Introduction to
Algebraic Geometry and Commutative Algebra, 2nd ed.* New York: Springer-Verlag, 1996.

Eisenbud, D. *Commutative Algebra with a View toward Algebraic Geometry.*
New York: Springer-Verlag, 1995.

Mishra, B. *Algorithmic Algebra.* New York: Springer-Verlag, 1993.

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1999-05-25