Cubic Thue inequalities with negative discriminant.
Let \(X^3+aX+b\) with \(a,b \in \mathbb Z\) be irreducible over \(\mathbb Z [X]\) with discriminant \(D\), and let \(\theta\) denote the real root (of smallest absolute value, if \(D>0\)) of this polynomial. Theorem 1 gives explicit expressions \(c(a,b), \lambda(a,b) > 0\) such that for all \(p,q \in \mathbb Z\) with \(q>0\) one has \[ \left| \theta- \frac pq \right| > \frac {c(a,b)}{q^{\lambda(a,b)}}, \] where \(\lambda(a,b)<3\) for \(| a| \) sufficiently large and tends to \(2\) for \(| a| \to \infty\). For the proof the author constructs suitable Padé approximations by using the integrals of \textit{J. H. Rickert} [Math. Proc. Camb. Philos. Soc. 113, 461--472 (1993; Zbl 0786.11040)]. This result is applied to give explicit bounds for the solutions of the Thue inequality \[ | x^3+axy^2+by^3| \leq k, \] supposing that \(a\) exceeds some positive bound depending on \(| b| \). As an application, for \(k=a+| b| +1\) the above inequality is completely solved for \(| b| \leq 2\) and arbitrary \(a>0\) as well as for all \((a,b)\) satisfying \(a \geq 360 b^4\). For this purpose, the first 21 terms of the continued fraction expansion of \(\theta\) are explicitly given.
scientific article; zbMATH DE number 1891440
scientific article