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Gevrey series of arithmetic type. II: Transcendence without transcendence. (Séries Gevrey de type arithmétique. II: Transcendance sans transcendance.) (French) Zbl 1037.11050
Summary: In this second part, we study the Diophantine properties of values of arithmetic Gevrey series of non-zero order at algebraic points. We rely on the fact, proved in the first part [ibid. 151, 705–740 (2000; Zbl 1037.11049)], that the minimal differential operator (with polynomial coefficients) which annihilates such a series has no non-trivial singularity outside the origin and infinity.
We show how to draw from this fact some transcendence properties and recover in particular the fundamental theorem of the Siegel-Shidlovsky theory on algebraic independence of values of $$E$$-functions. The paradox of the title points out the contrast between the qualitative aspect of this new argument and the essentially quantitative aspect of the traditional approach. Finally we discuss $$q$$-analogues of the theory (theta-functions, $$q$$-exponential,…).

MSC:
 11J81 Transcendence (general theory) 34M99 Ordinary differential equations in the complex domain
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