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Effective results for unit equations over finitely generated integral domains

Published online by Cambridge University Press:  23 November 2012

JAN–HENDRIK EVERTSE  and
KÁLMÁN GYŐRY
JAN–HENDRIK EVERTSE
Affiliation:
Leiden University, Mathematical Institute, P.O. Box 9512, 2300 RA Leiden, The Netherlands. e-mail: [email protected]
KÁLMÁN GYŐRY
Affiliation:
Institute of Mathematics, University of Debrecen, Number Theory Research Group, Hungarian Academy of Sciences, P.O. Box 12, H-4010 Debrecen, Hungary. e-mail: [email protected]
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Abstract

Let A ⊃ ℤ be an integral domain which is finitely generated over ℤ and let a,b,c be non-zero elements of A. Extending earlier work of Siegel, Mahler and Parry, in 1960 Lang proved that the equation (*) aϵ +bη = c in ϵ, η ∈ A* has only finitely many solutions. Using Baker's theory of logarithmic forms, Győry proved, in 1979, that the solutions of (*) can be determined effectively if A is contained in an algebraic number field. In this paper we prove, in a quantitative form, an effective finiteness result for equations (*) over an arbitrary integral domain A of characteristic 0 which is finitely generated over ℤ. Our main tools are already existing effective finiteness results for (*) over number fields and function fields, an effective specialization argument developed by Győry in the 1980's, effective results of Hermann (1926) and Seidenberg (1974) on linear equations over polynomial rings over fields, and similar such results by Aschenbrenner, from 2004, on linear equations over polynomial rings over ℤ. We prove also an effective result for the exponential equation aγ1v1···γsvs+bγ1w1 ··· γsws=c in integers v1,…,ws, where a,b,c and γ1,…,γs are non-zero elements of A.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 2 , March 2013 , pp. 351 - 380
Copyright
Copyright © Cambridge Philosophical Society 2012

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