S-unit

Not to be confused with S units used to measure signal strength with an S meter.

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 field with ring of integers R. Let S be a finite set of prime ideals of R. An element x of K is an S-unit if the principal fractional ideal (x) is a product of primes in S (to positive or negative powers). For the ring of rational integers Z one may take S to be a finite set of prime numbers and define an S-unit to be a rational number whose numerator and denominator are divisible only by the primes in S.

Properties

The S-units form a multiplicative group containing the units of R.

Dirichlet's unit theorem holds for S-units: the group of S-units is finitely generated, with rank (maximal number of multiplicatively independent elements) equal to r + s, where r is the rank of the unit group and s = |S|.

S-unit equation

The S-unit equation is a Diophantine equation

u + v = 1

with u and v restricted to being S-units of K (or more generally, elements of a finitely generated subgroup of the multiplicative group of any field of characteristic zero). The number of solutions of this equation is finite^[1] and the solutions are effectively determined using estimates for linear forms in logarithms as developed in transcendental number theory. A variety of Diophantine equations are reducible in principle to some form of the S-unit equation: a notable example is Siegel's theorem on integral points on elliptic curves, and more generally superelliptic curves of the form yⁿ = f(x).

A computational solver for S-unit equation is available in the software SageMath.^[2]

References

1. Beukers, F.; Schlickewei, H. (1996). "The equation x+y=1 in finitely generated groups". Acta Arithmetica. 78 (2): 189–199. doi:10.4064/aa-78-2-189-199. ISSN 0065-1036.
2. "Solve S-unit equation x + y = 1 — Sage Reference Manual v8.7: Algebraic Numbers and Number Fields". doc.sagemath.org. Retrieved 2019-04-16.

• Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs. Vol. 104. Providence, RI: American Mathematical Society. pp. 19–22. ISBN 0-8218-3387-1. Zbl 1033.11006.
• Lang, Serge (1978). Elliptic curves: Diophantine analysis. Grundlehren der mathematischen Wissenschaften. Vol. 231. Springer-Verlag. pp. 128–153. ISBN 3-540-08489-4.
• Lang, Serge (1986). Algebraic number theory. Springer-Verlag. ISBN 0-387-94225-4. Chap. V.
• Smart, Nigel (1998). The algorithmic resolution of Diophantine equations. London Mathematical Society Student Texts. Vol. 41. Cambridge University Press. Chap. 9. ISBN 0-521-64156-X.
• Neukirch, Jürgen (1986). Class field theory. Grundlehren der mathematischen Wissenschaften. Vol. 280. Springer-Verlag. pp. 72–73. ISBN 3-540-15251-2.

Further reading

• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge University Press. ISBN 978-0-521-88268-2.
• Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1130.11034.