AF+BG theorem


AF+BG theorem

In algebraic geometry, a field of mathematics, the AF+BG 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.

This theorem provides an analog for polynomials of Bézout'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 the greatest 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 P2. Assume that the curves determined by "F" and "G" have no common irreducible components, 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 the local ring of P2 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
.


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Theorem — The o*rem, n. [L. theorema, Gr. ? a sight, speculation, theory, theorem, fr. ? to look at, ? a spectator: cf. F. th[ e]or[ e]me. See {Theory}.] 1. That which is considered and established as a principle; hence, sometimes, a rule. [1913 Webster]… …   The Collaborative International Dictionary of English

  • Theorem of Pappus — Theorem The o*rem, n. [L. theorema, Gr. ? a sight, speculation, theory, theorem, fr. ? to look at, ? a spectator: cf. F. th[ e]or[ e]me. See {Theory}.] 1. That which is considered and established as a principle; hence, sometimes, a rule. [1913… …   The Collaborative International Dictionary of English

  • Theorem der korrespondierenden Zustände — Theorem der korrespondierenden Zustände,   Theorem der übereinstimmenden Zustände, 1881 von J. D. van der Waals aufgestellter thermodynamischer Satz, der davon ausgeht, dass sich alle Stoffe am kritischen Punkt (kritischer Zustand) in einem… …   Universal-Lexikon

  • Theorem Stencil — Theorem stencil, sometimes also called theorem painting or velvet painting, is the art of making stencils and using them to make drawings or paintings on fabric or paper. [Chotner, Deborah (1992). American Naive Paintings , pp. 370 71. Oxford… …   Wikipedia

  • Theorem — The o*rem, v. t. To formulate into a theorem. [1913 Webster] …   The Collaborative International Dictionary of English

  • Theorem Proving in Higher-Order Logics — (TPHOLs) is an annual international academic conference on the topic of automated reasoning in higher order logics. The first TPHOLs was held in Cambridge, UK in 1987, but in the early years was an informal gathering of researchers interested in… …   Wikipedia

  • Theorēm — (griech.), soviel wie Lehrsatz …   Meyers Großes Konversations-Lexikon

  • Theorem — Theorēm (grch.), Lehrsatz …   Kleines Konversations-Lexikon

  • Theorem — Theorem, griech. (das Wort mit allen damit verwandten Ausdrücken hat seine Wurzel in einem griech. Zeitworte, welches bedeutet: anschauen, besonders: innerlich, geistig anschauen, in seinen letzten Gründen etwas schauen), der Lehrsatz; t. atisch …   Herders Conversations-Lexikon

  • theorem — index inference, postulate, prescription (directive), principle (axiom), supposition Burton s Legal Thesaurus. William …   Law dictionary

  • Theorem — Theorem,das:⇨Lehrsatz …   Das Wörterbuch der Synonyme