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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.
See also Buchberger's Algorithm, Commutative Algebra
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.