Linear forms in the values of $G$-functions, and Diophantine equations

E. M. Matveev

Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 379-396

Résumé

Using a rather general theorem on $G$-functions proved in this paper, the author establishes the existence of an effective upper bound for the solutions of certain Diophantine equations, such as those of the form $$ a_1x^g_1-a_2x^g_2=p_1^{z_1}\cdots p_k^{z_k}G(x_1,x_2), $$ where $a_1,a_2$ and $p_1,\dots,p_k$ are natural numbers and $G(x_1, x_2)$ is a polynomial of small degree. The upper bound has the form $$ \max(|x_1|,|x_2|)\leqslant(\xi H(G))^{1/(g-\gamma-\operatorname{deg}G)}, $$ where $\gamma$ depends on $a_1,a_2$ and $p_1,\dots,p_k$ and can be written out explicitly, and $\xi$ is an effective positive constant. Bibliography: 17 titles.