On Thue inequalities and a conjecture of Schmidt
Let \(F[X,Y] \in \mathbb{Z} [X,Y]\) be a binary form of degree \(d\), which is irreducible over \(\mathbb{Q}\). Let \(m \in \mathbb{N}\) and consider the number \(N_F (m)\) of solutions of the Thue inequality \(|F(x,y) |\leq m\) in \(x,y, \in \mathbb{Z}\). In 1987 \textit{W. Schmidt} [Trans. Am. Math. Soc. 303, 241-255 (1987; Zbl 0634.10017)] showed that for forms of degree \(d \geq 3\) with \(s + 1\) nonzero coefficients \[ N_F(m) \ll \sqrt {ds} \cdot m^{2/d} (1 + \log m^{1/d}), \] where the implicit constant is absolute. Schmidt also conjectured that here the logarithmic term was unnecessary. This was recently shown to be true for \(d = 3\) by the author [Acta Arith. 66, No. 3, 237-243 (1994; Zbl 0807.11018)]. In the present paper the author proves that Schmidt's conjecture is true for \(d > 3\), at least for \(m\) sufficiently large, namely for \[ \bigl |D(F) \bigr |^{1/d(d - 1)} \leq \log \log m. \] The proof combines ideas of the above mentioned result of Schmidt, an explicit version of Mahler's result due to \textit{M. Bean} [Bounds for the number of solutions of Thue equations, M. thesis, Univ. of Waterloo (1988)], and estimates of \textit{J. Evertse} and \textit{K. Györy} [Compos. Math. 79, 169-204 (1991; Zbl 0746.11020)].
scientific article; zbMATH DE number 770126