- AF+BG theorem
In

algebraic geometry , a field ofmathematics , the**AF+BG theorem**is a result ofMax Noether which describes when the equation of analgebraic curve in thecomplex projective plane can be written in terms of the equations of two other algebraic curves.This theorem provides an analog for

polynomial s ofBézout'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 thegreatest 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".**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

**P**^{2}. Assume that the curves determined by "F" and "G" have no commonirreducible component s, 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 thelocal ring of**P**^{2}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".**References***citation

last1 = Griffiths | first1 = Phillip

last2 = Harris | first2 = Joseph

isbn = 9780471050599

publisher = John Wiley & Sons

title = Principles of Algebraic Geometry

year = 1978.

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