AF+BG theorem

Let F, G, and H be homogeneous polynomials in three variables, with H having higher degree than F and G; let a = deg H − deg F and b = deg H − deg G (both positive integers) be the differences of the degrees of the polynomials. Suppose that the greatest common divisor of F and G is a constant, which means that the projective curves that they define in the projective plane ⁠${\displaystyle \mathbb {P} ^{2}}$⁠ have an intersection consisting in a finite number of points. For each point P of this intersection, the polynomials F and G generate an ideal (F, G)_P of the local ring of ⁠${\displaystyle \mathbb {P} ^{2}}$⁠ at P (this local ring is the ring of the fractions ⁠${\displaystyle {\tfrac {n}{d}},}$⁠ where n and d are polynomials in three variables and d(P) ≠ 0). The theorem asserts that, if H lies in (F, G)_P for every intersection point P, then H lies in the ideal (F, G); that is, there are homogeneous polynomials A and B of degrees a and b, respectively, such that H = AF + BG. Furthermore, any two choices of A differ by a multiple of G, and similarly any two choices of B differ by a multiple of F.