Siegel identity

In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.

Statement

The first formula is

${\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}=1.}$

The second is

${\displaystyle {\frac {x_{3}-x_{1}}{x_{2}-x_{1}}}\cdot {\frac {t-x_{2}}{t-x_{3}}}+{\frac {x_{2}-x_{3}}{x_{2}-x_{1}}}\cdot {\frac {t-x_{1}}{t-x_{3}}}=1.}$

Application

The identities are used in translating Diophantine problems connected with integral points on hyperelliptic curves into S-unit equations.

See also

• Siegel formula

References

• Baker, Alan (1975). Transcendental Number Theory. Cambridge University Press. p. 40. ISBN 0-521-20461-5. Zbl 0297.10013.
• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. 9. Cambridge University Press. p. 53. ISBN 978-0-521-88268-2. Zbl 1145.11004.
• Kubert, Daniel S.; Lang, Serge (1981). Modular Units. Grundlehren der Mathematischen Wissenschaften. 244. ISBN 0-387-90517-0.
• Lang, Serge (1978). Elliptic Curves: Diophantine Analysis. Grundlehren der mathematischen Wissenschaften. 231. Springer-Verlag. ISBN 0-387-08489-4.
• Smart, N. P. (1998). The Algorithmic Resolution of Diophantine Equations. London Mathematical Society Student Texts. 41. Cambridge University Press. pp. 36–37. ISBN 0-521-64633-2.