AF+BG theorem

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In algebraic geometry, a field of mathematics, the AF+BG theorem (also known as Max Noether's fundamental 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.


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 constant, which means that the projective curves that they define in the projective plane P2 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 P2 at P (this local ring is the ring of the fractions 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 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.

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This theorem may be viewed as a generalization of Bézout's identity, which provides a condition under which an integer or a univariate polynomial h may be expressed as an element of the ideal generated by two other integers or univariate polynomials 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 homogeneous polynomial H in three variables can be written as an element of the ideal generated by two other polynomials F and G.

This theorem is also a refinement, for this particular case, of Hilbert's Nullstellensatz, which provides a condition expressing that some power of a polynomial h (in any number of variables) belongs to the ideal generated by a finite set of polynomials.


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