S\\éries Gevrey de type arithm\\étique, II. Transcendance sans transcendance

By Y. Andr'e. 2000

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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, 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. At last, we discuss q-analogues of the theory (theta-functions, q-exponential,...).

Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science

View Paper: https://www.semanticscholar.org/paper/180403654eeae3804771f9228ce0c3405b2ceba3