Buchberger's algorithm

The Euclidean algorithm for computing the polynomial greatest common divisor is a special case of Buchberger's algorithm restricted to polynomials of a single variable. Gaussian elimination of a system of linear equations is another special case where the degree of all polynomials equals one.

For other Gröbner basis algorithms, see Gröbner basis § Algorithms and implementations.

Algorithm

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

Input A set of polynomials F that generates I

Output A Gröbner basis G for I

1. G := F
2. For every f_i, f_j in G, denote by g_i the leading term of f_i with respect to the given monomial ordering, and by a_ij the least common multiple of g_i and g_j.
3. Choose two polynomials in G and let S_ij = ⁠a_ij/ g_i⁠ f_i − ⁠a_ij/ g_j⁠ f_j (The leading terms here will cancel by construction).
4. Reduce S_ij, with the multivariate division algorithm relative to the set G until the result is not further reducible. If the result is non-zero, add it to G.
5. Repeat steps 2–4 until all possible pairs are considered, including those involving the new polynomials added in step 4.
6. Output G

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 many ways to improve this algorithm beyond what has been stated above. For example, all the new elements of F can be reduced 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 using only f_i and f_j for reduction), so it needn't be calculated 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.