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Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions

Abstract : The Siegel-Shidlovskii Theorem states that the transcendence degree of the field generated over Q(z) by E-functions solutions of a differential system of order 1 is the same as the transcendence degree of the field generated over Q by the evaluation of these E-functions at non-zero algebraic points (expect possibly at a finite number of them). The analogue of this theorem is false for G-functions and we present conditional and unconditional results showing that any intermediate numerical transcendence degree can be obtained.
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Tanguy Rivoal. Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2015, 38, pp.29-35. ⟨hal-01253639⟩

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