Quantitative problems on the size of \(G\)-operators (Q2162760)

A \(G\)-function is a power series satisfying certain hypotheses. Examples of \(G\)-functions are certain hypergeometric series with rational parameters, polylogarithms and the function \(1/(1-z)\). \(G\)-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's \(G\)-functions, satisfy the Galochkin condition of moderate growth, encoded by a \(p\)-adic quantity, the size. Previous works of Chudnovsky, André and Dwork have provided inequalities between the size of a \(G\)-operator and certain computable constants depending among others on its solutions. The author recalls André's idea to attach a notion of size to differential modules and details his results on the behavior of the size relatively to the standard algebraic operations on the modules. This allows to prove a quantitative version of André's generalization of Chudnovsky's Theorem: for \(f(z)=\Sigma _{\alpha ,k,\ell}c_{\alpha ,k,\ell}\log ^k(z)f_{\alpha ,k,\ell}(z)\), where \(f_{\alpha ,k,\ell}\) are \(G\)-functions, one can determine an upper bound on the size of the minimal operator \(L\) over \(\bar{\mathbb{Q}}(z)\) of \(f(z)\) in terms of quantities depending on the \(f_{\alpha ,k,\ell}\), the rationals \(\alpha\) and the integers \(k\). The author gives two applications of this result: an estimation of the size of a product of two \(G\)-operators depending on the size of each operator and a computation of a constant appearing in a certain Diophantine problem.