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Siegel's Theorem
In mathematics, Siegel's theorem on integral points states that for a Singular point of an algebraic variety, smooth algebraic curve ''C'' of genus (mathematics), genus ''g'' defined over a number field ''K'', presented in affine space in a given coordinate system, there are only finitely many points on ''C'' with coordinates in the ring of integers ''O'' of ''K'', provided ''g'' > 0. The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations. For ''g'' > 1 it was superseded by Faltings's theorem in 1983. History In 1929, Siegel proved the theorem by combining a version of the Thue–Siegel–Roth theorem, from diophantine approximation, with the Mordell–Weil theorem from diophantine geometry (required in Weil's version, to apply to the Jacobian variety of ''C''). In 2002, Umberto Zannier and Pietro Corvaja gave a new proof by using a new me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
