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On Runge’s theorem about diophantine equations. (English) Zbl 0849.11033

Halász, G. (ed.) et al., Sets, graphs and numbers. A birthday salute to Vera T. Sós and András Hajnal. Amsterdam: North-Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 60, 329-356 (1992).
Runge’s theorem to which the title refers [C. Runge, Über ganzzahlige Lösungen von Gleichungen zwischen zwei Veränderlichen, J. Reine Angew. Math. 100, 425–435 (1887; JFM 19.0076.03)] claims that if \(f(x, y)=0\) has infinitely many solutions \((x, y)\in \mathbb Z^2\), where \(f\in \mathbb Z [X, Y]\) is irreducible, then four explicitly given conditions on \(f\) must hold, the first of which is that the highest powers of \(X\) and \(Y\) in \(f\) occur as isolated terms \(aX^m\) and \(bY^n\). One of the most significant features of this theorem is that if these conditions are not fulfilled, all integral solutions of \(f(x, y)=0\) may be effectively determined.
In the present paper the authors give explicit upper bounds for integers \(|x|\) and \(|y|\) for which \(f(x, y)= 0\) in terms of \(m= \deg_X f\), \(n= \deg_Y f\) and the height \(|f|\) when one of the Runge conditions does not hold. For instance, in case the first condition (see above) is not satisfied \[ \max \{|x|, |y|^n\}\leq \bigl( (m+1) (n+1) (mn+1 )^{2/n} |f|)^{2n( mn+ 1)^3}. \]
For the entire collection see [Zbl 0925.05001].

MSC:

11D41 Higher degree equations; Fermat’s equation

Citations:

JFM 19.0076.03
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