AF+BG theorem

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In algebraic geometry, a field of mathematics, the AF+BG theorem is a result of Max Noether which describes when the equation of an algebraic curve in the complex projective plane can be written in terms of the equations of two other algebraic curves.

This theorem provides an analog for polynomials of Bézout's identity, which provides a condition under which an integer h can be written as a sum of integer multiples of two other integers f and g: such a representation exists exactly when h is a multiple of the greatest common divisor of f and g. The AF+BG condition expresses, in terms of divisors (sets of points, with multiplicities), a similar condition under which a polynomial H can be written as a sum of polynomial multiples of two other polynomials F and G.

[edit] Statement of the theorem

Let F, G, and H be homogenous polynomials in three variables, and assume that a = deg H − deg F and b = deg H − deg G are positive integers. The points where F, G, and H are zero determine algebraic curves in the projective plane P². Assume that the curves determined by F and G have no common irreducible components, meaning that the intersection of F and G is a set of points. At every point P in this intersection, F and G generate an ideal (F, G)_P of the local ring of P² at P; we assume that H always lies in (F, G)_P. The theorem then says that there are homogenous 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.