Shuji Morikawa, Hiroshi Umemura (2009)
Annales de l’institut Fourier
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We apply the General Galois Theory of difference equations introduced in the first part to concrete examples. The General Galois Theory allows us to define a discrete dynamical system being infinitesimally solvable, which is a finer notion than being integrable. We determine all the infinitesimally solvable discrete dynamical systems on the compact Riemann surfaces.
Pierre Dèbes, Nour Ghazi (2012)
Annales de l’institut Fourier
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Our main result combines three topics: it contains a Grunwald-Wang type conclusion, a version of Hilbert’s irreducibility theorem and a -adic form à la Harbater, but with good reduction, of the Regular Inverse Galois Problem. As a consequence we obtain a statement that questions the RIGP over . The general strategy is to study and exploit the good reduction of certain twisted models of the covers and of the associated moduli spaces.
Childs, Lindsay N. (2011)
The New York Journal of Mathematics [electronic only]
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Guy Casale (2009)
Annales de l’institut Fourier
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In this article we give an obstruction to integrability by quadratures of an ordinary differential equation on the differential Galois group of variational equations of any order along a particular solution. In Hamiltonian situation the condition on the Galois group gives Morales-Ramis-Simó theorem. The main tools used are Malgrange pseudogroup of a vector field and Artin approximation theorem.
Moshe Jarden (2006)
Journal de Théorie des Nombres de Bordeaux
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We prove that if is a number field and is a Galois extension of which is not algebraically closed, then is not PAC over .