Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Linear independence of values of G-functions, II. Outside the disk of
  convergence   Given any non-polynomial $G$-function $F(z)=\sum_{k=0}^\infty A_k z^k$ of
radius of convergence $R$ and in the kernel a $G$-operator $L_F$, we consider
the $G$-functions $F_n^{[s]}(z)=\sum_{k=0}^\infty \frac{A_k}{(k+n)^s}z^k$ for
every integers $s\ge 0$ and $n\ge 1$. These functions can be analytically
continued to a domain $\mathcal{D}_F$ star-shaped at $0$ and containing the
disk $\{\vert z\vert <R\}$. Fix any $\alpha \in\mathcal D_F \cap
\overline{\mathbb{Q}}^*$, not a singularity of $L_F$, and any number field
$\mathbb{K}$ containing $\alpha$ and the $A_k$'s. Let $\Phi_{\alpha, S}$ be the
$\mathbb{K}$-vector space spanned by the values $F_n^{[s]}(\alpha)$, $n\ge 1$
and $0\le s\le S$. We prove that $u_{\mathbb{K},F}\log(S)\le
\dim_\mathbb{K}(\Phi_{\alpha, S })\le v_FS$ for any $S$, for some constants
$u_{\mathbb{K},F}>0$ and $v_F>0$. This appears to be the first Diophantine
result for values of $G$-functions evaluated outside their disk of convergence.
This theorem encompasses a previous result of the authors in [{\em Linear
independence of values of G-functions}, 46 pages, J. Europ. Math. Soc., to
appear], where $\alpha\in \overline{\mathbb{Q}}^*$ was assumed to be such that
$\vert \alpha\vert <R$. Its proof relies on an explicit construction of a
Pad\'e approximation problem adapted to certain non-holomorphic functions
associated to $F$, and it is quite different of that in the above mentioned
paper. It makes use of results of Andr\'e, Chudnovsky and Katz on
$G$-operators, of a linear independence criterion \`a la Siegel over number
fields, and of a far reaching generalization of Shidlovsky's lemma built upon
the approach of Bertrand-Beukers and Bertrand.

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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: cs