Classification of algebraic solutions of irregular Garnier systems

We prove that algebraic solutions of Garnier systems in the irregular case are of two types. The classical ones come from isomonodromic deformations of linear equations with diagonal or dihedral differential Galois group; we give a complete list in the rank-2 case (two indeterminates). The pull-back ones come from deformations of coverings over a fixed degenerate hypergeometric equation; we provide a complete list when the differential Galois group is $\text{SL}_{2}(\mathbb{C})$. As a byproduct, we obtain a complete list of algebraic solutions for the rank-2 irregular Garnier systems.

Complex Dynamics of Some Hamiltonian Systems: Nonintegrability of Equations of Motion

The main purpose of this paper is to study the complexity of some Hamiltonian systems from the view of nonintegrability, including the planar Hamiltonian with Nelson potential, double-well potential, and the perturbed elliptic oscillators Hamiltonian. Some numerical analyses show that the dynamic behavior of these systems is very complex and in fact chaotic in a large range of their parameter. I prove that these Hamiltonian systems are nonintegrable in the sense of Liouville. My proof is based on the analysis of normal variational equations along some particular solutions and the investigation of their differential Galois group.

Algebraic solutions of differential equations over ℙ1 −{0,1,∞}

The Grothendieck–Katz [Formula: see text]-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo [Formula: see text] has vanishing [Formula: see text]-curvatures for almost all [Formula: see text] has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on [Formula: see text] We prove a variant of this conjecture for [Formula: see text] which asserts that if the equation satisfies a certain convergence condition for all [Formula: see text] then its monodromy is trivial. For those [Formula: see text] for which the [Formula: see text]-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of [Formula: see text]-curvatures and certain local monodromy groups. We also prove similar variants of the [Formula: see text]-curvature conjecture for an elliptic curve with [Formula: see text]-invariant [Formula: see text] minus its identity and for [Formula: see text].

Reproduction of Fuchs Relation for the Group $S_4^{SL_2}$ by Using Groebner Bases

In 1872, Lazarus Fuchs used a new tool which is The Invariant Theory to construct the minimal polynomial of an algebraic solution of a differential equation of second order. He expressed the coefficient of the equation in terms of the (semi-)invariants of its differential Galois group. In this paper we will give another method to obtain Fuchs Relation: for the octahedral groupe $S_4^{SL_2}$ by using Groebner Basis; a tool which is introduced in 1965 nearly two century after Fuchs Relation.

Calculating Galois groups of third-order linear differential equations with parameters

Motivated by developing algorithms that decide hypertranscendence of solutions of extensions of the Bessel differential equation, algorithms computing the unipotent radical of a parameterized differential Galois group have been recently developed. Extensions of Bessel’s equation, such as the Lommel equation, can be viewed as homogeneous parameterized linear differential equations of the third order. In this paper, we give the first known algorithm that calculates the differential Galois group of a third-order parameterized linear differential equation.

Computation of the difference-differential Galois group and differential relations among solutions for a second-order linear difference equation

We apply the difference-differential Galois theory developed by Hardouin and Singer to compute the differential-algebraic relations among the solutions to a second-order homogeneous linear difference equation [Formula: see text] where the coefficients [Formula: see text] are rational functions in [Formula: see text] with coefficients in [Formula: see text]. We develop algorithms to compute the difference-differential Galois group associated to such an equation, and show how to deduce the differential-algebraic relations among the solutions from the defining equations of the Galois group.