Groebner Bases

The study of ideals in polynomial rings over fields is absolutely central to algebraic geometry. We think of an affine variety (usually called affine algebraic set without an irreducibility hypothesis) in n dimensions as defined to be the common zero set of a collection of polynomials f1(x1, . . . , xn), . . . , fm(x1, . . . , xn) with coefficients in some field k. We see that if a point (c1, . . . , cn) is a zero of each of the fi, then it is a zero of any polynomials in the ideal I ⊆ k[x1, . . . , xn] generated by the xi. We further see that if f ∈ k[x1, . . . , xn] has f ∈ I for some d ≥ 1, then f(c1, . . . , cn) = 0, so f(c1, . . . , cn) = 0. This motivates the following definitions: Definition 1.1. Given an ideal I ⊆ k[x1, . . . , xn], we define V (I) ⊆ k to be the set of points (c1, . . . , cn) with f(c1, . . . , cn) = 0 for all f ∈ I. Given a subset S ⊆ k, we define I(S) ⊆ k[x1, . . . , xn] to be the set of functions f such that f(c1, . . . , cn) = 0 for all (c1, . . . , cn) ∈ S. Finally, if I ⊆ R is an ideal in a commutative ring, we set rad(I) := {f ∈ R : ∃d ≥ 1, f ∈ I}, and say I is radical if rad(I) = I.