Groebner Bases and Applications

f = gq1 + r1 g = r1q2 + r2 r1 = r2q3 + r3 . . . rn−2 = rn−1qn + rn rn−1 = rnqn+1 + 0 Furthermore, rn = af + bg for explicitly computable a, b ∈ k[x] (solving the equations above). We can use these algorithms to decide things such like ideal membership (when is f ∈ (f1, . . . , fm)) and equality (when does (f1, . . . , fm) = (g1, . . . , gl)). In the above, we used the degree of a polynomial as a measure of the size of a polynomial and the algorithms eventually terminate by producing polynomials of lesser degree at each step. To extend these ideas to polynomials in several variables we need a notion of size for polynomials (with nice properties).