Differential Galois Theory and Integration

Differential Galois Theory and Isomonodromic Deformations

We present a geometric setting for the differential Galois theory of $G$-invariant connections with parameters. As an application of some classical results on differential algebraic groups and Lie algebra bundles, we see that the Galois group of a connection with parameters with simple structural group $G$ is determined by its isomonodromic deformations. This allows us to compute the Galois groups with parameters of the general Fuchsian special linear system and of Gauss hypergeometric equation.

The inverse problem of differential Galois theory over the field R(z)

We describe a Picard-Vessiot theory for differential fields with non algebraically closed fields of constants. As a technique for constructing and classifying Picard-Vessiot extensions, we develop a Galois descent theory. We utilize this theory to prove that every linear algebraic group $G$ over $\mathbb{R}$ occurs as a differential Galois group over $\mathbb{R}(z)$. The main ingredient of the proof is the Riemann-Hilbert correspondence for regular singular differential equations over $\mathbb{C}(z)$.

Introduction to the Galois Theory of Linear Differential Equations

This is an expanded version of the 10 lectures given as the 2006 London Mathematical Society Invited Lecture Series at the Heriot-Watt University 31 July - 4 August 2006.

Computing the differential Galois group of a parameterized second-order linear differential equation

We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form \[ \tfrac{\partial^2}{\partial x^2} Y + r_1 \tfrac{\partial}{\partial x} Y + r_0 Y = 0, \] where the coefficients $r_1, r_0 \in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under the assumption that $r_1 = 0$. We show how to complete this procedure to cover the cases where $r_1 \neq 0$, by reinterp…

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.

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 the present paper, we give the first known algorithm that calculates the differential Galois group of a third order parameterized linear differential equation.

Lie's Reduction Method and Differential Galois Theory in the Complex Analytic Context

This paper is dedicated to the differential Galois theory in the complex analytic context for Lie-Vessiot systems. Those are the natural generaliza- tion of linear systems, and the more general class of differential equations adimitting superposition laws, as recently stated in [5]. A Lie-Vessiot sys- tem is automatically translated into a equation in a Lie group that we call automorphic system. Reciprocally an automorphic system induces a hierarchy of Lie-Vessiot systems. In this work we study the global analytic aspects of a classical method of reduction of differential equations, due to S. …

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without par…

Differential Galois Theory of Algebraic Lie-Vessiot Systems

In this paper we develop a differential Galois theory for algebraic Lie-Vessiot systems in algebraic homogeneous spaces. Lie-Vessiot systems are non autonomous vector fields that are linear combinations with time-dependent coefficients of fundamental vector fields of an algebraic Lie group action. Those systems are the building blocks for differential equations that admit superposition of solutions. Lie-Vessiot systems in algebraic homogeneous spaces include the case of linear differential equations. Therefore, the differential Galois theory for Lie-Vessiot systems is an extension of the class…

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…

正交微分系统与微分伽罗瓦理论的达布变换

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

本文提出一种将达布变换在算法上推广至线性微分系统张量积构造的方法，获得了三阶正交系统的显式变换及扩展至SL(2,C)-系统任意对称幂的框架，并在Maple中验证。

Reductive linear differential algebraic groups and the Galois groups of parameterized linear differential equations

We develop the representation theory for reductive linear differential algebraic groups (LDAGs). In particular, we exhibit an explicit sharp upper bound for orders of derivatives in differential representations of reductive LDAGs, extending existing results, which were obtained for SL(2) in the case of just one derivation. As an application of the above bound, we develop an algorithm that tests whether the parameterized differential Galois group of a system of linear differential equations is reductive and, if it is, calculates it.

On the Constructive Inverse Problem in Differential Galois Theory

We give sufficient conditions for a linear differential equation to have a given semisimple group as its Galois group. For any linear algebraic group G given as a semidirect product of a finite subgroup and a normal subgroup that is a product of groups of type An, Cn, Dn, E6, or E7, we construct a differential equation over C(x) having Galois group G.

Linear differential equations with finite differential Galois group

For a differential operator $L$ of order $n$ over $C(z)$ with a finite (differential) Galois group $G\subset {\rm GL}(C^n)$, there is an algorithm, by M. van Hoeij and J.-A.~Weil, which computes the associated evaluation of the invariants $ev:C[X_1,\dots ,X_n]^G\rightarrow C(z)$. The procedure proposed here does the opposite: it uses a theorem of E.~Compoint and computes the operator $L$ from a given evaluation $h$. Moreover it solves a part of the inverse problem of producing $L$ for a given representation of a finite group $G$. Another part considered here, is finding irreducible $G$-invaria…

Approche galoisienne de la transcendance différentielle

In this survey we present the parameterized Galois theory of difference equations, as introduced by Hardouin-Singer. The purpose of this theory is to give a systematic approach to differential transcendence, also called hypertranscendence. As an example of the applications, we explain the Galoisian proof of Hölder's theorem on the differential transcendence of the Gamma function.

A Galoisian proof of Ritt theorem on the differential transcendence of Poincaré functions

Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincaré" by J.F. Ritt, on the differential transcendence of the solutions of the functional equation R(y(t))=y(qt), where R is a rational function with complex coefficients which verifies R(0)=0, R'(0)=q, where q is a complex number with |q|>1. We also give a partial result in the case of an algebraic function R.

Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation

We propose a new method to compute the unipotent radical $R_u(H)$ of the differential Galois group $H$ associated to a parameterized second-order homogeneous linear differential equation of the form \[\tfrac{\partial^2}{\partial x^2}Y-qY=0,\] where $q \in F(x)$ is a rational function in $x$ with coefficients in a $Π$-field $F$ of characteristic zero, and $Π$ is a commuting set of parametric derivations. The procedure developed by Dreyfus reduces the computation of $R_u(H)$ to solving a creative telescoping problem, whose effective solution requires the assumption that the maximal reductive quo…

Computing the differential Galois group of a one-parameter family of second order linear differential equations

We develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider equations of the form \frac{\partial^2Y}{\partial x^2}+ r_1\frac{\partial Y}{\partial x} +r_2Y=0, where $r_1,r_2\in C(x,t)$ and $C$ is an algebraically closed field of characteristic zero. We work in the setting of parameterized Picard-Vessiot theory, which attaches a linear differential algebraic group to such an equation, that is, a group of invertible matrices…

三阶微分方程的Liouvillian解

Liouvillian Solutions of Third Order Differential Equations

本文针对三阶线性微分方程设计算法计算Liouvillian解，通过优化不变量计算微分Galois群，并在可解情形以广义超几何函数的拉回和规范变换表示解，计算效率实用。

Projective Isomonodromy and Galois Groups

In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic condition for a paramaterized linear differential equation to be projectively isomonodromic, in terms of the derived group of its parameterized Picard-Vessiot group.

关于有限微分 Galois 群逆问题的一个注记

A note on the inverse problem for finite differential Galois groups

本文提出一种算法，在给定商曲线函数域时，构造常线性微分方程使其 Schwarz 映射参数化不可约有限线性代数群下的不变曲线，推广了已有逆问题解，并证明了不存在微分 Galois 群同构于 $H_{72}^{SL_3}$ 的超几何方程。

A Galois-theoretic proof of the differential transcendence of the incomplete Gamma function

We give simple necessary and sufficient conditions for the $\frac{\partial}{\partial t}$-transcendence of the solutions to a parameterized second order linear differential equation of the form \frac{\partial^2 Y}{\partial x^2} - p \frac{\partial Y}{\partial x} = 0, where $p\in F(x)$ is a rational function in $x$ with coefficients in a $\frac{\partial}{\partial t}$-field $F$. This result is crucial for the development of an efficient algorithm to compute the parameterized Picard-Vessiot group of an arbitrary parameterized second-order linear differential equation over $F(x)$. Our criteria imply…

Linear Algebraic Groups as Parameterized Picard-Vessiot Galois Groups

We show that a linear algebraic group is the Galois group of a parameterized Picard-Vessiot extension of k(x), x' = 1, for certain differential fields k, if and only if its identity component has no one dimensional quotient as a linear algebraic group.

贝尔数及相关序列的微分超越性：一种伽罗瓦理论方法

Differential transcendence of Bell numbers and relatives: a Galois theoretic approach

本研究利用差分伽罗瓦理论证明了生成函数微分超越性的一般性结论，将其应用于多类生成函数（含贝尔数等Sheffer类示例），为Pak-Yeliussizov猜想提供了具体证据。

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。

Reversible linear differential equations

Let $\nabla$ be a meromorphic connection on a vector bundle over a compact Riemann surface $Γ$. An automorphism $σ:Γ\toΓ$ is called a symmetry of $\nabla$ if the pull-back bundle and the pull-back connection can be identified with $\nabla$. We study the symmetries by means of what we call the Fano Group; and, under the hypothesis that $\nabla$ has a unimodular reductive Galois group, we relate the differential Galois group, the Fano group and the symmetries by means of an exact sequence.

Some tastings in Morales-Ramis Theory

In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian Systems through Liouville Arnold Theorem and Integrability of Linear Differential Equations through Differential Galois Theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system,

Isomonodromic differential equations and differential categories

We study isomonodromicity of systems of parameterized linear differential equations and related conjugacy properties of linear differential algebraic groups by means of differential categories. We prove that isomonodromicity is equivalent to isomonodromicity with respect to each parameter separately under a filtered-linearly closed assumption on the field of functions of parameters. Our result implies that one does not need to solve any non-linear differential equations to test isomonodromicity anymore. This result cannot be further strengthened by weakening the requirement on the parameters a…