Quantitative problems on the size of $G$-operators

$G$-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's $G$-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a $p$-adic quantity, the size. Previous works of Chudnovsky, Andr\'e and Dwork have provided inequalities between the size of a $G$ -operator and certain computable constants depending among others on its solutions. First, we recall Andr\'e's idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of Andr\'e's generalization of Chudnovsky's Theorem: for $f(z)=\sum_{\alpha, k,\ell} c_{\alpha, k,\ell} z^{\alpha} \log(z)^k f_{\alpha, k,\ell}(z)$, where $f_{\alpha, k,\ell}(z)$ are $G$-functions, we can determine an upper bound on the size of the minimal operator $L$ over $\overline{\mathbb{Q}}(z)$ of $f(z)$ in terms of quantities depending on the $f_{\alpha, k,\ell}(z)$, the rationals $\alpha$ and the integers $k$. We give two applications of this result: we estimate the size of a product of two $G$-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author.