Representability of G-functions as rational functions in hypergeometric series

Fres\'an and Jossen have given a negative answer to a question of Siegel about the representability of every $E$-function as a polynomial with algebraic coefficients in $E$-functions of type ${}_pF_q[\underline{a};\underline{b};\gamma x^{q-p+1}]$ with $q\geq p\geq 0$, $\gamma \in \overline{\mathbb Q}$ and rational parameters $\underline{a}, \underline{b}$. In this paper, we study, in a more general context, a similar question for $G$-functions asked by Fischler and the second author: can every $G$-function be represented as a polynomial with algebraic coefficients in $G$-functions of type $\mu(x)\cdot {}_pF_{p-1}[\underline{a};\underline{b};\lambda(x)]$ with $p\ge 1$, rational parameters $\underline{a},\underline{b}$ and $\mu,\lambda$ algebraic over $\mathbb Q(x)$ with $\lambda(0)=0$? They have shown the answer to be negative under a generalization of Grothendieck's Period Conjecture and a technical assumption on the~$\lambda$'s. Using differential Galois theory, we prove that, for every $N\in \mathbb N$, there exists a $G$-function which can not be represented as a rational function with coefficients in $\overline{\mathbb C(x)}$ of solutions of linear differential equations with coefficients in $\mathbb C(x)$ and at most $N$ singularities in $\mathbb{P}^1 (\mathbb C)$. As a corollary, we deduce that not all $G$-functions can be represented as a rational function in hypergeometric series of the above mentioned type, when the $\lambda$'s are rational functions with degrees of their numerators and denominators bounded by an arbitrarily large fixed constant. This provides an unconditional negative answer to the question asked by Fischler and the second author for such~$\lambda$'s.