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AF+BG theorem
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.
== Statement==
Let ''F'', ''G'', and ''H'' be homogeneous polynomials in three variables, such that ''a'' = deg ''H'' − deg ''F'' and ''b'' = deg ''H'' − deg ''G'' are positive integers. We 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''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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