AF+BG theorem

In algebraic geometry the AF+BG theorem (also known as Max Noether's fundamental theorem) is a result of Max Noether that asserts that, if the equation of an algebraic curve in the complex projective plane belongs locally (at each intersection point) to the ideal generated by the equations of two other algebraic curves, then it belongs globally to this ideal.

Statement

Let Template:Mvar, Template:Mvar, and Template:Mvar be homogeneous polynomials in three variables, with Template:Mvar having higher degree than Template:Mvar and Template:Mvar; 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 Template:Mvar and Template:Mvar is a constant, which means that the projective curves that they define in the projective plane Template:Tmath have an intersection consisting in a finite number of points. For each point Template:Mvar of this intersection, the polynomials Template:Mvar and Template:Mvar generate an ideal (F, G)_P of the local ring of Template:Tmath at Template:Mvar (this local ring is the ring of the fractions Template:Tmath where Template:Mvar and Template:Mvar are polynomials in three variables and d(P) ≠ 0). The theorem asserts that, if Template:Mvar lies in (F, G)_P for every intersection point Template:Mvar, then Template:Mvar lies in the ideal (F, G); that is, there are homogeneous polynomials Template:Mvar and Template:Mvar of degrees Template:Mvar and Template:Mvar, respectively, such that H = AF + BG. Furthermore, any two choices of Template:Mvar differ by a multiple of Template:Mvar, and similarly any two choices of Template:Mvar differ by a multiple of Template:Mvar.

Related results

This theorem may be viewed as a generalization of Bézout's identity, which provides a condition under which an integer or a univariate polynomial Template:Mvar may be expressed as an element of the ideal generated by two other integers or univariate polynomials Template:Mvar and Template:Mvar: such a representation exists exactly when Template:Mvar is a multiple of the greatest common divisor of Template:Mvar and Template:Mvar. The AF+BG condition expresses, in terms of divisors (sets of points, with multiplicities), a similar condition under which a homogeneous polynomial Template:Mvar in three variables can be written as an element of the ideal generated by two other polynomials Template:Mvar and Template:Mvar.

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 Template:Mvar (in any number of variables) belongs to the ideal generated by a finite set of polynomials.

References

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