An ideal in the ring F[x, y] is defined as any set of bivariate polynomials that satisfies a certain pair of closure conditions. Examples of ideals can arise in several ways. The most direct way to specify concretely an ideal in the ring F[x, y] is by giving a set of generator polynomials. The ideal is then the set of all polynomial combinations of the generator polynomials. These generator polynomials need not necessarily form a minimal basis. We may wish to compute a minimal basis for an ideal by starting with a given set of generator polynomials. We shall describe an algorithm, known as the Buchberger algorithm, for this computation. Thus, given a set of generator polynomials for an ideal, the Buchberger algorithm computes another set of generator polynomials for that ideal that is a minimal basis.

A different way of specifying an ideal in the ring F[x, y] is as a locator ideal for the nonzeros of a given bivariate polynomial. We then may wish to express this ideal in terms of a set of generator polynomials for it, preferably a set of minimal polynomials. Again, we need a way to compute a minimal basis, but starting now from a different specification of the ideal. We shall describe an algorithm, known as the Sakata algorithm, that performs this computation.