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The 4-value theorem in number theory. (Der 4-Werte-Satz in der Zahlentheorie.) (German) Zbl 0784.11010
The author remarks that just as it takes four points \(a_ i\) in \(\mathbb{C}\) so that coincidence of the divisors \({\mathcal D}(f-a_ i)={\mathcal D}(g-a_ i)\) guarantees the equality of the meromorphic functions \(f\) and \(g\), so it requires four distinct elements \(\alpha_ i\) of a number field so that the equalities \({\mathcal D}(\varphi- \alpha_ i)={\mathcal D}(\gamma- \alpha_ i)\) be satisfied by just finitely many pairs \((\varphi,\gamma)\) of elements of the field. Of course now \({\mathcal D}(\beta)\) means \(\sum_{\mathfrak p} \text{ord}_{\mathfrak p} \beta{\mathfrak p}\) with the sum running over the prime ideals of the field. The vital tool in this argument is the \(S\)-unit theorem of A. J. van der Poorten and H. P. Schlickewei [J. Aust. Math. Soc., Ser. A 51, 154-170 (1991; Zbl 0747.11017)] and of J.-H. Evertse [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008)].

MSC:
11D61 Exponential Diophantine equations
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
11R99 Algebraic number theory: global fields
Keywords:
divisor; \(S\)-units
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References:
[1] Evertse, J.M. : On sums of S-units and linear recurrences . Compos. Math. 53, 225-244 (1984). · Zbl 0547.10008 · numdam:CM_1984__53_2_225_0 · eudml:89685
[2] Langmann, K. : Anwendungen des Satzes von Picard . Math. Ann. 266, 369-390 (1984). · Zbl 0578.32051 · doi:10.1007/BF01475586 · eudml:163862
[3] Laurent, M. : Equations diophantiennes exponentielles . Invent. Math. 78, 299-327 (1984). · Zbl 0554.10009 · doi:10.1007/BF01388597 · eudml:143175
[4] Nevanlinna, R. : Le théorème de Picard-Borel et la théorie des functions méromorphs . New York: Chelsea 1974. · Zbl 0357.30019
[5] Van Der Poorten, A.J. and Schlickewei, H.P. : The growth conditions for recurrence sequences . Macquarie Math. Rep. 82-0041; North-Ryde, Australia (1982).
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