An algorithm for computing a standard form for second-order linear q-difference equations

In this article an algorithm is presented for computing a standard form for second order linear q-difference equations. This standard form is useful for determining the q-difference Galois group and the set of Liouvillian solutions of a given equation. (C) 1997 Elsevier Science B.V.</p

Differential transcendence criteria for second-order linear difference equations and elliptic hypergeometric functions

We develop general criteria that ensure that any non-zero solution of a given second-order difference equation is differentially transcendental, which apply uniformly in particular cases of interest, such as shift difference equations, q-dilation difference equations, Mahler difference equations, and elliptic difference equations. These criteria are obtained as an application of differential Galois theory for difference equations. We apply our criteria to prove a new result to the effect that most elliptic hypergeometric functions are differentially transcendental

Differential Galois Approach to the Non-integrability of the Heavy Top Problem

We study integrability of the Euler-Poisson equations describing the motion of a rigid body with one fixed point in a constant gravity field. Using the Morales-Ramis theory and tools of differential algebra we prove that a symmetric heavy top is integrable only in the classical cases of Euler, Lagrange, and Kovalevskaya and is partially integrable only in the Goryachev-Chaplygin case. Our proof is alternative to that given by Ziglin ({\em Funktsional. Anal. i Prilozhen.}, 17(1):8--23, 1983; {\em Funktsional. Anal. i Prilozhen.}, 31(1):3--11, 95, 1997).Comment: 31 pages, 1 figur

Nonintegrability of the two-body problem in constant curvature spaces

We consider the reduced two-body problem with the Newton and the oscillator potentials on the sphere ${\bf S}^{2}$ and the hyperbolic plane ${\bf H}^{2}$. For both types of interaction we prove the nonexistence of an additional meromorphic integral for the complexified dynamic systems.Comment: 20 pages, typos correcte