Differential transcendency in the theory of linear differential systems with constant coefficients

Linear Differential Systems with Small Coefficients: Various Types of Solvability and their Verification

We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have sufficiently small coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated…

On commuting ordinary differential operators with polynomial coefficients corresponding to spectral curves of genus two

The group of automorphisms of the first Weyl algebra acts on commuting ordinary differential operators with polynomial coefficient. In this paper we prove that for fixed generic spectral curve of genus two the set of orbits is infinite.

化简具有一般系数的矩阵微分方程

Simplifying matrix differential equations with general coefficients

本文利用微分 Galois 理论与微分本质维数，证明含一般系数的矩阵微分方程经有理规范变换后至少仍需 n 个参数，并给出一般 Picard-Vessiot 扩张参数数的界。

Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients

In this paper we construct examples of commuting ordinary scalar differential operators with polynomial coefficients that are related to a spectral curve of an arbitrary genus g>0 and to an arbitrary rank r>1 of the vector bundle of common eigenfunctions of the commuting operators over the spectral curve. This solves completely the well-known existence problem for commuting operators of arbitrary genus and arbitrary rank with polynomial coefficients. The constructed commuting operators of arbitrary rank r>1 and arbitrary genus g>0 are given explicitly, they are generated by the Chebyshev polyn…