In
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 ...
, Siegel's identity refers to one of two formulae that are used in the resolution of
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
s.
Statement
The first formula is
:
The second is
:
Application
The identities are used in translating Diophantine problems connected with integral points on
hyperelliptic curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form
y^2 + h(x)y = f(x)
where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dist ...
s into
S-unit equation
In mathematics, in the field of algebraic number theory, an ''S''-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for ''S''-units.
Definition
Let ''K'' be a number ...
s.
See also
*
Siegel formula
References
*
*
*
*
* {{cite book , title=The Algorithmic Resolution of Diophantine Equations , volume=41 , series=London Mathematical Society Student Texts , first=N. P. , last=Smart , authorlink=Nigel Smart (cryptographer) , publisher=
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press
A university press is an academic publishing hou ...
, year=1998 , isbn=0-521-64633-2 , page
36–37, url=https://archive.org/details/algorithmicresol0000smar/page/36
Algebraic identities
Diophantine equations