Quasialgebraicity of Picard--Vessiot fields

Skolem-Mahler-Lech type theorems and Picard-Vessiot theory

We show that three problems involving linear difference equations with rational function coefficients are essentially equivalent. The first problem is the generalization of the classical Skolem-Mahler-Lech theorem to rational function coefficients. The second problem is the question whether or not for a given linear difference equation there exists a Picard-Vessiot extension inside the ring of sequences. The third problem is a certain special case of the dynamical Mordell-Lang conjecture. This allows us to deduce solutions to the first two problems in a particular but fairly general special ca…

On the integrability of polynomial fields in the plane by means of Picard-Vessiot theory

We study the integrability of polynomial vector fields using Galois theory of linear differential equations when the associated foliations is reduced to a Riccati type foliation. In particular we obtain integrability results for some families of quadratic vector fields, Liénard equations and equations related with special functions such as Hypergeometric and Heun ones. The Poincaré problem for some families is also approached.

微分方程的Liouvillian首次积分：一种Galois方法

Liouvillian first integrals of differential equations: A Galoisian approach

利用微分Galois理论证明一阶微分方程组存在Liouvillian首次积分等价于其在Liouvillian Picard-Vessiot扩张或环中存在，给出Singer定理的新证明，并展示Liouvillian假设不可去掉。

A note about rational representations of differential Galois groups

We give a description of the rational representations of the differential Galois group of a Picard-Vessiot extension.

Integrability conditions at order 2 for homogeneous potentials of degree -1

We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1. Conversely, we prove that if this criterion is satisfied, then the Galois group of second order variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.

Unipotent differential algebraic groups as parameterized differential Galois groups

We deal with aspects of the direct and inverse problems in parameterized Picard-Vessiot (PPV) theory. It is known that, for certain fields, a linear differential algebraic group (LDAG) G is a PPV Galois group over these fields if and only if G contains a Kolchin-dense finitely generated group. We show that, for a class of LDAGs G, including unipotent groups, G is such a group if and only if it has differential type 0. We give a procedure to determine if a parameterized linear differential equation has a PPV Galois group in this class and show how one can calculate the PPV Galois group of a par…