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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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