Английская Википедия:AF+BG theorem
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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 Шаблон:Mvar, Шаблон:Mvar, and Шаблон:Mvar be homogeneous polynomials in three variables, with Шаблон:Mvar having higher degree than Шаблон:Mvar and Шаблон:Mvar; let Шаблон:Math and Шаблон:Math (both positive integers) be the differences of the degrees of the polynomials. Suppose that the greatest common divisor of Шаблон:Mvar and Шаблон:Mvar is a constant, which means that the projective curves that they define in the projective plane Шаблон:Tmath have an intersection consisting in a finite number of points. For each point Шаблон:Mvar of this intersection, the polynomials Шаблон:Mvar and Шаблон:Mvar generate an ideal Шаблон:Math of the local ring of Шаблон:Tmath at Шаблон:Mvar (this local ring is the ring of the fractions Шаблон:Tmath where Шаблон:Mvar and Шаблон:Mvar are polynomials in three variables and Шаблон:Math). The theorem asserts that, if Шаблон:Mvar lies in Шаблон:Math for every intersection point Шаблон:Mvar, then Шаблон:Mvar lies in the ideal Шаблон:Math; that is, there are homogeneous polynomials Шаблон:Mvar and Шаблон:Mvar of degrees Шаблон:Mvar and Шаблон:Mvar, respectively, such that Шаблон:Math. Furthermore, any two choices of Шаблон:Mvar differ by a multiple of Шаблон:Mvar, and similarly any two choices of Шаблон:Mvar differ by a multiple of Шаблон: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 Шаблон:Mvar may be expressed as an element of the ideal generated by two other integers or univariate polynomials Шаблон:Mvar and Шаблон:Mvar: such a representation exists exactly when Шаблон:Mvar is a multiple of the greatest common divisor of Шаблон:Mvar and Шаблон:Mvar. The AF+BG condition expresses, in terms of divisors (sets of points, with multiplicities), a similar condition under which a homogeneous polynomial Шаблон:Mvar in three variables can be written as an element of the ideal generated by two other polynomials Шаблон:Mvar and Шаблон: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 Шаблон:Mvar (in any number of variables) belongs to the ideal generated by a finite set of polynomials.
References
External links
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