Ideal Membership Problem

The Ideal Membership Problem is as follows: given f0, f1, . . . , fm ∈ K[x1, . . . , xn], is f0 ∈ 〈f1, . . . , fm〉, where 〈f1, . . . , fm〉 denotes the ideal generated by the fi? An equivalent formulation is: are there q1, . . . , qm ∈ K[x1, . . . , xn] such that f0 = ∑m i=1 qifi? We will solve this question by using Gröbner bases. That is, a Gröbner basis is a “nice” representation of an ideal, that allows us to easily decide membership. The difficult part of the analysis is to construct Gröbner bases, and we will do so using Buchberger’s algorithm (Buchberger’s work essentially started the field of Gröbner bases, and he named the notion after his advisor, Gröbner). The Gröbner basis technique has turned out to be fairly successful in practice, although its theoretical guarantees are quite weak (and provably so). However, it is still a nice theory, and in particular unifies two otherwise disparate topics: solving linear systems of equations, and computing greatest common divisors of univariate polynomials. We now discuss these two relations. Suppose the polynomials fi are all linear with no constant term, and we still want to solve the ideal membership question. One can then show that it is enough to assume the qi are in fact constants from the field K, instead of general polynomials in K[x1, . . . , xn]. Thus, in this case the ideal membership question is just that of solving a linear system, which can be solved by Gaussian elimination. In this case, the notion of a Gröbner basis in fact reduces to the notion of row-reduced echelon form. Further, recall that solving a linear system is quite easy given the row-reduced echelon form. Constructing the echelon form can be done via Gaussian Elimination, which is comparably more expensive. The same notions will be true of Gröbner bases. In another simple case of ideal membership, suppose we have all univariate polynomials, so n = 1. In this case, we can recall that 〈f1, . . . , fm〉 = 〈gcd(f1, . . . , fm)〉. So to test if f0 ∈ 〈f1, . . . , fm〉, it suffices to test if gcd(f1, . . . , fm) divides f0. Constructing this gcd seems the comparably more expensive part of this test, and once given the gcd the division is quite quick. Now that we have discussed two subcases of this question, let us consider the entire ideal membership question. The starting point is that given f0, we want to compute its “remainder” modulo the fi. This requires some notion of division for multivariate polynomials.