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Exceptional values of E ‐functions at algebraic points
Author(s) -
Adamczewski Boris,
Rivoal Tanguy
Publication year - 2018
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12168
Subject(s) - transcendental function , algebraic function , mathematics , algebraic number , algebraic extension , transcendental number , algebraic element , function field of an algebraic variety , real algebraic geometry , algebraic cycle , function (biology) , diophantine approximation , differential algebraic geometry , entire function , dimension of an algebraic variety , exponential function , differential (mechanical device) , singular point of an algebraic variety , algebraic surface , algebraic differential equation , diophantine equation , discrete mathematics , pure mathematics , differential equation , ordinary differential equation , mathematical analysis , differential algebraic equation , aerospace engineering , biology , engineering , evolutionary biology
E ‐functions are entire functions with algebraic Taylor coefficients satisfying certain arithmetic conditions, and which are also solutions of linear differential equations with coefficients inQ ¯ ( z ) . They were introduced by Siegel in 1929 to generalize Diophantine properties of the exponential function, and studied further by Shidlovskii in 1956. The celebrated Siegel–Shidlovskii theorem deals with the algebraic (in)dependence of values at algebraic points of E ‐functions solutions of a differential system. However, somewhat paradoxically, this deep result may fail to decide whether a given E ‐function assumes an algebraic or a transcendental value at some given algebraic point. Building upon André's theory of E ‐operators, Beukers refined in 2006 the Siegel–Shidlovskii theorem in an optimal way. In this paper, we use Beukers' work to prove the following result: there exists an algorithm which, given a transcendental E ‐function f ( z ) as input, outputs the finite list of all exceptional algebraic points α such that f ( α ) is also algebraic, together with the corresponding list of values f ( α ) . This result solves the problem of deciding whether values of E ‐functions at algebraic points are transcendental.