Siegel identity explained

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

(x _{3} - x _{1})/(x _{2} - x _{1}) + (x _{2} - x _{3})/(x _{2} - x _{1}) = 1 .

The second is

(x _{3} - x _{1})/(⋅ x _{2} - x _{1}) (t - x _{2})/(t - x _{3}) + (x _{2} - x _{3})/(x _{2} - x _{1}) ⋅ (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

• Book: Baker, Alan . Alan Baker (mathematician)

. Alan Baker (mathematician) . Transcendental Number Theory . . 1975 . 0-521-20461-5 . 0297.10013 . 40 .

• Book: Alan . Baker . Alan Baker (mathematician). Gisbert . Wüstholz . Gisbert Wüstholz . Logarithmic Forms and Diophantine Geometry . New Mathematical Monographs . 9 . . 2007 . 978-0-521-88268-2 . 1145.11004 . 53 .
• Book: Daniel S. . Kubert . Daniel Kubert . Serge . Lang . Serge Lang . Modular Units . Grundlehren der Mathematischen Wissenschaften . 244 . 1981 . 0-387-90517-0 .
• Book: Lang, Serge . Serge Lang

. Serge Lang . Elliptic Curves: Diophantine Analysis . 231 . Grundlehren der mathematischen Wissenschaften . . 1978 . 0-387-08489-4 .

• Book: Smart, N. P. . Nigel Smart (cryptographer)

. The Algorithmic Resolution of Diophantine Equations . 41 . London Mathematical Society Student Texts . Nigel Smart (cryptographer) . . 1998 . 0-521-64633-2 . 36–37 .