Buchberger's algorithm

In computational algebraic geometry, 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 ordering.

A crude version of this algorithm proceeds as follows:

1. Start with F={f₁, f₂,...,f_k}, a set of generators for your ideal. Let g_i be the leading term of f_i with respect to the given ordering, and denote the greatest common divisor of g_i and g_j by a_ij.

2. Let S_ij=(g_j /a_ij) f_i-(g_i /a_ij) f_j. Note that the leading terms here will cancel by construction.

3. Using the multivariate division algorithm, reduce all the S_ij relative to the set F.

4. Add all the nonzero resultants of step 3 to F, and repeat steps 1-4 until nothing new is added.

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. It also should be noted that if the leading terms of f_i and f_j share no variables in common, then S_ij will always reduce to 0, so we needn't calculate it at all.

We are 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. Therefore this algorithm does indeed stop. Unfortunately, it may take a very long time to terminate, corresponding to the fact that Gröbner bases can be extremely large.