On the solutions of Thue equations
Let \(k\) be a number field and let \({\mathfrak o}_k\) denote its ring of integers. Let \(p(X,Y) \in {\mathfrak o}_k [X,Y]\) be a homogeneous polynomial of degree \(n \geq 3\), and let \(a \in {\mathfrak o}_k\). The Thue equation is the equation \(p(X,Y) = a\), to be solved with \(X,Y \in {\mathfrak o}_k\). The present paper finds upper bounds on the number of solutions of a restricted type of Thue equation in terms of the rank \(R_a\) of the group of \(k\)-rational points of the Jacobian of the associated projective curve \(p(X,Y) = Z^n\). More precisely, write \((a) = {\mathfrak a} {\mathfrak b}^n\), where \({\mathfrak a}\) is an ideal in \({\mathfrak o}_k\) not divisible by the \(n\)-th power of any prime ideal, and \({\mathfrak b}\) is an ideal. Let \(e(a) = n \log |N^k_\mathbb{Q} {\mathfrak b} |/ \log |N^k_\mathbb{Q} {\mathfrak a} |\). The paper shows that if \(n > 3\), if \(p(X,Y)\) has a linear factor in \(k[X,Y]\), if \(e(a) \leq 1\), and if \(|N^k_\mathbb{Q} a |\) is sufficiently large, then the number of solutions of the Thue equation \(p(X,Y) = a\) is at most \(4 \cdot 7^{R_a}\). More generally, this is proved for \(S\)-integers. This refines a result of \textit{J. H. Silverman} [Invent. Math. 74, 281-292 (1983; Zbl 0525.14012)]. Both papers use the results of \textit{D. Mumford} [Am. J. Math. 87, 1007-1016 (1965; Zbl 0151.273)]; the present paper adds the contribution of \textit{E. Bombieri} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, 615-640 (1990; Zbl 0722.14010); errata-corrige, ibid. 18, 473 (1991; Zbl 0763.14007)]. Other bounds on the number of solutions of the Thue equation are proved, e.g., by \textit{J.-H. Evertse} and \textit{K. Györy} [J. Reine Angew. Math. 358, 6-19 (1985; Zbl 0552.10010)].
scientific article; zbMATH DE number 729876
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