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Inalgebraic geometry theAF+BG theorem (also known asMax Noether's fundamental theorem) is a result ofMax Noether that asserts that, if the equation of analgebraic curve in thecomplex projective plane belongs locally (at each intersection point) to theideal generated by the equations of two other algebraic curves, then it belongs globally to this ideal.

LetF,G, andH behomogeneous polynomials in three variables, withH having higher degree thanF andG; leta = degH − degF andb = degH − degG (both positive integers) be the differences of the degrees of the polynomials. Suppose that thegreatest common divisor ofF andG is a constant, which means that theprojective curves that they define in the projective plane⁠${\displaystyle {\mathbb{P}}^2}$⁠ have an intersection consisting in a finite number of points. For each pointP of this intersection, the polynomialsF andG generate anideal(F,G)_P of thelocal ring of⁠${\displaystyle {\mathbb{P}}^2}$⁠ atP (this local ring is the ring of the fractions⁠${\displaystyle \frac{n}{d},}$⁠ wheren andd are polynomials in three variables andd(P) ≠ 0). The theorem asserts that, ifH lies in(F,G)_P for every intersection pointP, thenH lies in the ideal(F,G); that is, there are homogeneous polynomialsA andB of degreesa andb, respectively, such thatH =AF +BG. Furthermore, any two choices ofA differ by a multiple ofG, and similarly any two choices ofB differ by a multiple ofF.

This theorem may be viewed as a generalization ofBézout's identity, which provides a condition under which an integer or a univariate polynomialh may be expressed as an element of theideal generated by two other integers or univariate polynomialsf andg: such a representation exists exactly whenh is a multiple of thegreatest common divisor off andg. The AF+BG condition expresses, in terms ofdivisors (sets of points, with multiplicities), a similar condition under which ahomogeneous polynomialH in three variables can be written as an element of the ideal generated by two other polynomialsF andG.

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

• Fulton, William (2008), "5.5 Max Noether's Fundamental Theorem and 5.6 Applications of Noether's Theorem",Algebraic Curves: An Introduction to Algebraic Geometry(PDF), pp. 60–65.
• Griffiths, Phillip; Harris, Joseph (1978),Principles of Algebraic Geometry, John Wiley & Sons,ISBN 978-0-471-05059-9.

• Weisstein, Eric W.,"Noether's Fundamental Theorem",MathWorld

• Theorems in algebraic geometry
• Theorems in complex geometry

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