A note about rational representations of differential Galois groups

Dynamics of rational symplectic mappings and difference Galois theory

In this paper we study the relationship between the integrability of rational symplectic maps and difference Galois theory. We present a Galoisian condition, of Morales-Ramis type, ensuring the non-integrability of a rational symplectic map in the non-commutative sense (Mishchenko-Fomenko). As a particular case, we obtain a com- plete discrete analogue of Morales-Ramis Theorems for non-integrabi- lity in the sense of Liouville.

具有非实 Julia 集的有理映射的树状 Galois 群

Arboreal Galois groups of rational maps with nonreal Julia sets

本文研究具有非实 Julia 集的有理映射的树状 Galois 群，证明相关映射及与椭圆曲线倍乘映射关联的 Lattès 映射具有非交换树状 Galois 群，并探讨实系数多项式何时具有实 Julia 集。

A Galois-dynamics correspondence for unicritical polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton-Patel, and Vivaldi-Hatjispyros in the early 1990s con…

具有碰撞临界点的二次有理函数的树状 Galois 群

Arboreal Galois groups for quadratic rational functions with colliding critical points

本文研究了二次有理函数临界点碰撞时的树状 Galois 群，给出了子群 $M_{\ell}$ 的新描述与 Pink 定理的新证明，并证明了 $G_{\infty}$ 成为全群的充要条件。

Differential Galois Theory and Lie Symmetries

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.