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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: evertse@math.leidenuniv.nl
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: gyory@science.unideb.hu
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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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References

REFERENCES

[1]Aschenbrenner, M.Ideal membership in polynomial rings over the integers. J. Amer. Math. Soc. 17 (2004), 407–442.CrossRefGoogle Scholar
[2]Baker, A.Contributions to the theory of Diophantine equations. Philos. Trans. Roy. Soc. London, Ser. A 263 (1968), 173–208.Google Scholar
[3]Bombieri, E. and Gubler, W.Heights in Diophantine Geometry (Cambridge University Press, 2006).Google Scholar
[4]Borosh, I., Flahive, M., Rubin, D. and Treybig, B.A sharp bound for solutions of linear Diophantine equations. Proc. Amer. Math. Soc. 105 (1989), 844–846.CrossRefGoogle Scholar
[5]Coates, J.An effective p-adic analogue of a theorem of Thue. Acta Arith. 15 (1968/69), 279–305.CrossRefGoogle Scholar
[6]Győry, K.Sur les polynômes à coefficients entiers et de discriminant donné II. Publ. Math. Debrecen 21 (1974), 125–144.CrossRefGoogle Scholar
[7]Győry, K.On the number of solutions of linear equations in units of an algebraic number field. Comment. Math. Helv. 54 (1979), 583–600.CrossRefGoogle Scholar
[8]Győry, K.Bounds for the solutions of norm form, discriminant form and index form equations in finitely generated domains. Acta Math. Hung. 42 (1983), 45–80.CrossRefGoogle Scholar
[9]Győry, K.Effective finiteness theorems for polynomials with given discriminant and integral elements with given discriminant over finitely generated domains. J. Reine Angew. Math. 346 (1984), 54–100.Google Scholar
[10]Győry, K. and Yu, KunruiBounds for the solutions of S-unit equations and decomposable form equations. Acta Arith. 123 (2006), 9–41.CrossRefGoogle Scholar
[11]Hartshorne, R.Algebraic Geometry (Springer Verlag, 1977).CrossRefGoogle Scholar
[12]Hermann, G.Die Frage der endlich vielen Schritte in der Theorie der Polynomideale. Math. Ann. 95 (1926), 736–788.CrossRefGoogle Scholar
[13]Lang, S.Integral points on curves. Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27–43.CrossRefGoogle Scholar
[14]Lewis, D.J. and Mahler, K.On the representation of integers by binary forms. Acta Arith. 6 (1961), 333–363.CrossRefGoogle Scholar
[15]Loher, T. and Masser, D.Uniformly counting points of bounded height. Acta Arith. 111 (2004), 277–297.CrossRefGoogle Scholar
[16]Louboutin, S.Explicit bounds for residues of dedekind zeta functions, values of L-functions at s=1, and relative class numbers. J. Number Theory 85 (2000), 263–282.CrossRefGoogle Scholar
[17]Loxton, J. H. and van der Poorten, A. J.Multiplicative dependence in number fields. Acta Arith. 42 (1983), 291–302.CrossRefGoogle Scholar
[18]Mahler, K.Zur Approximation algebraischer Zahlen, I. (Über den größten Primteiler binärer Formen) Math. Ann. 107 (1933), 691–730.CrossRefGoogle Scholar
[19]Mason, R. C.The hyperelliptic equation over function fields. Math. Proc. Camb. Phil. Soc. 93 (1983), 219–230.CrossRefGoogle Scholar
[20]Parry, C. J.The -adic generalisation of the Thue-Siegel theorem. Acta Math. 83 (1950), 1–100.CrossRefGoogle Scholar
[21]van der Poorten, A. J. and Schlickewei, H. P.Additive relations in fields. J. Austral. Math. Soc. (Ser. A) 51 (1991), 154–170.CrossRefGoogle Scholar
[22]Roquette, P.Einheiten und Divisorenklassen in endlich erzeugbaren Körpern. Jber. Deutsch. Math. Verein 60 (1958), 1–21.Google Scholar
[23]Schmidt, W. M.Thue's equation over function fields. J. Austral. Math. Soc. Ser. A 25 (1978), 385–422.CrossRefGoogle Scholar
[24]Seidenberg, A.Constructions in algebra. Trans. Amer. Math. Soc. 197 (1974), 273–313.CrossRefGoogle Scholar
[25]Siegel, C. L.Approximation algebraischer Zahlen. Math. Zeitschrift 10 (1921), 173–213.CrossRefGoogle Scholar
[26]Simmons, H.The solution of a decision problem for several classes of rings. Pacific J. Math. 34 (1970), 547–557.CrossRefGoogle Scholar