On the resolution of Thue inequalities
The author describes a practical method to find all solutions to the inequality \[ | F(x,y)| \leq m\qquad\text{in}\;x,y\in\mathbb Z \tag{*} \] with \((x,y)=1\), where \(F(X,Y)\in\mathbb Z[X,Y]\) is a homogeneous polynomial of degree \(n\geq 3\) which is irreducible over \(\mathbb Q[X,Y]\) and \(m\) is a positive integer. Using a result of \textit{A. Baker} [Philos. Trans. R. Soc. Lond., Ser. A 263, 173--191 (1968; Zbl 0157.09702)] one can compute a number \(y_ 0\), depending on \(F\) and \(m\), such that \(| y| \leq y_ 0\) for each solution \((x,y)\) to (*). Unfortunately, this bound \(y_ 0\) is too large to be of any practical help in the resolution of (*). The author develops a method to reduce a given upper bound \(y_ 0\) for \(| y|\) to a much smaller upper bound for \(| y|\). His method uses the continued fraction expansions of the zeros of \(F(x,1)\): assuming that the partial quotients of the zeros of \(F(x,1)\) grow at most polynomially whereas the denominators of their convergents grow exponentially, the author's method reduces the bound \(y_ 0\) to a bound which is at most a power of \(\log y_ 0\). As an application of his method the author solves the inequality \[ | x^3+x^2y-2xy^2-y^3| \leq 200,\qquad | y| \leq 10^{500} \] in integers \(x,y\) with \((x,y)=1\).
scientific article; zbMATH DE number 4014804
scientific article