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

Tanguy Rivoal (1)

1 IF - Institut Fourier

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.

Mots clés: logarithmic singularity; G-functions; Siegel-Shidlovskii Theorem

Domaines: Mathématiques [math]

Tanguy Rivoal. Remarks on the impossibility of a Siegel-Shidlovskii like theorem for G-functions. Hardy-Ramanujan Journal, 2015, Volume 38 - 2015, pp.29-35. ⟨10.46298/hrj.2015.1357⟩. ⟨hal-01253639⟩