Geodesic
Linear forms in the values of $G$-functions, and Diophantine equations
Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 379-396 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{SM_1983_45_3_a3,
     author = {E. M. Matveev},
     title = {Linear forms in the values of $G$-functions, and {Diophantine} equations},
     journal = {Sbornik. Mathematics},
     pages = {379--396},
     year = {1983},
     volume = {45},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1983_45_3_a3/}
}
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E. M. Matveev. Linear forms in the values of $G$-functions, and Diophantine equations. Sbornik. Mathematics, Tome 45 (1983) no. 3, pp. 379-396. http://geodesic.mathdoc.fr/item/SM_1983_45_3_a3/

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