Buchberger's algorithm

In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. It was invented by Austrian mathematician Bruno Buchberger. One can view it as a generalization of the Euclidean algorithm for univariate GCD computation and of Gaussian elimination for linear systems.

A crude version of this algorithm to find a basis for an ideal I of a polynomial ring R proceeds as follows:

The polynomial S_ij is commonly referred to as the S-polynomial, where S refers to subtraction (Buchberger) or Syzygy (others). The pair of polynomials with which it is associated is commonly referred to as critical pair.

There are numerous ways to improve this algorithm beyond what has been stated above. For example, one could reduce all the new elements of F relative to each other before adding them. If the leading terms of f_i and f_j share no variables in common, then S_ij will always reduce to 0 (if we use only f_i and f_j for reduction), so we needn't calculate it at all.

The algorithm terminates because it is consistently increasing the size of the monomial ideal generated by the leading terms of our set F, and Dickson's lemma (or the Hilbert basis theorem) guarantees that any such ascending chain must eventually become constant.

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