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

Picard-Vessiot域的拟代数性

Quasialgebraicity of Picard--Vessiot fields

证明了在单值群的谱假设下，黎曼球面上的Fuchsian线性方程组的解的孤立零点个数有明确的整体界.

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…

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。