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On the algebraic dependence of E ‐functions
Author(s) -
Rivoal T.,
Roques J.
Publication year - 2016
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/bdw003
Subject(s) - mathematics , power series , algebraic number , differential (mechanical device) , linear differential equation , order (exchange) , taylor series , class (philosophy) , pure mathematics , series (stratigraphy) , differential equation , algebra over a field , algebraic differential equation , differential algebraic equation , mathematical analysis , ordinary differential equation , paleontology , finance , aerospace engineering , artificial intelligence , computer science , engineering , economics , biology
Siegel introduced and studied the class of E ‐functions in 1929. They are power series, solutions of some linear differential equations, whose Taylor coefficients satisfy certain arithmetic and growth conditions. After the work of Siegel and Shidlovskii, and its refinement by Beukers, on the algebraic relations between values of E ‐functions over Q ¯ , it is important to know when the E ‐functions solutions of a given differential system of order 1 are algebraically dependent or not overQ ¯ ( z ) . In this paper, we give the complete classification of the vector solutions of two‐dimensional differential systems of order 1 whose components are algebraically dependent E ‐functions overQ ¯ ( z ) .