We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.
Differential Galois theory has played important roles in the theory of integrability of linear differential equation. In this paper we will extend the theory to nonlinear case and study the integrability of the first order nonlinear differential equation. We will define for the differential equation the differential Galois group, will study the structure of the group, and will prove the equivalent between the existence of the Liouvillian first integral and the solvability of the corresponding differential Galois group.
We present a Galois theory of parameterized linear differential equations where the Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters and satisfy a set of differential equations with respect to these parameters. We present the basic constructions and results, give examples, discuss how isomonodromic families fit into this theory and show how results from the theory of linear differential algebraic groups may be used to classify systems of second order linear differential equations.
We extend Kovacic's algorithm to compute the differential Galois group of some second order parameterized linear differential equation. In the case where no Liouvillian solutions could be found, we give a necessary and sufficient condition for the integrability of the system. We give various examples of computation.
In this paper, we present methods to simplify reducible linear differential systems before solving. Classical integrals appear naturally as solutions of such systems. We will illustrate the methods developed in a previous paper on several examples to reduce the differential system. This will give information on potential algebraic relations between integrals.
We study parameterized linear differential equations with coefficients depending meromorphically upon the parameters. As a main result, analogously to the unparameterized density theorem of Ramis, we show that the parameterized monodromy, the parameterized exponential torus and the parameterized Stokes operators are topological generators in Kolchin topology, for the parameterized differential Galois group introduced by Cassidy and Singer. We prove an analogous result for the global parameterized differential Galois group, which generalizes a result by Mitschi and Singer. These authors give al…
The aim of this article is to provide a method to prove the irreducibility of non-linear ordinary differential equations by means of the differential Galois group of their variational equations along algebraic solutions. We show that if the dimension of the Galois group of a variational equation is large enough then the equation must be irreducible. We propose a method to compute this dimension via reduced forms. As an application, we reprove the irreducibility of the second and third Painlevé equations for special values of their parameter. In the Appendix, we recast the various notions of va…
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.
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 $\mathbb{C}(z)$.
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.
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…
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.
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.
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. …
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…
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…
This paper deals with criteria of algebraic independence for the derivatives of solutions of rank one difference equations. The key idea consists in deriving from the commutativity of the differentiation and difference operators a sequence of iterated extensions of the original difference module, thereby setting the problem in the framework of difference Galois theory and finally reducing it to an exercise in linear algebra. The involved tannakian categories are neutral over non necessarily algebraically closed fields, and this leads us to study the behaviour of Galois groups under base field …
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…
We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of Picard-Vessiot extensions over fields with not necessarily algebraically closed subfields of constants.
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.
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.
We apply Kovacic's algorithm from differential Galois theory to show that all complex non-oscillatory solutions (finite exponential of convergence of zeros) of certain Hill equations considered by Bank and Laine using Nevanlinna theory must be Liouvillian solutions. That is, solutions are obtainable by suitable differential field extensions construction. In particular, we have established a full correspondence between solutions of non-oscillatory type and Liouvillian solutions for a particular Hill equation. Explicit closed-form solutions are obtained via both methods for this Hill equation wh…
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…
In this paper we compute the difference Galois groups of the Lie-irreducible regular singular generalized q-hypergeometric equations of order 3 with q-real parameters by using a density theorem due to Sauloy. In contrast with the differential case, we show that these groups automatically contain the special linear group SL(3,C).
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.
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.
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…
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…
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.
We consider first-order linear difference systems over $\mathbb{C}(x)$, with respect to a difference operator $σ$ that is either a shift $σ:x\mapsto x+1$, $q$-dilation $σ:x\mapsto qx$ with $q\in{\mathbb{C}^\times}$ not a root of unity, or Mahler operator $σ:x\mapsto x^q$ with $q\in\mathbb{Z}_{\geq 2}$. Such a system is integrable if its solutions also satisfy a linear differential system; it is projectively integrable if it becomes integrable "after moding out by scalars." We apply recent results of Schäfke and Singer to characterize which groups can occur as Galois groups of integrable or pro…
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…
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.
Differential transcendence of Bell numbers and relatives: a Galois theoretic approach
Inria, Université Paris-Saclay（法国国家信息与自动化研究所，巴黎萨克雷大学）；CNRS, Université Paris-Saclay, UVSQ, Laboratoire de mathématiques de Versailles（法国国家科学研究中心，巴黎萨克雷大学，凡尔赛大学，凡尔赛数学实验室）；CNRS, Université Angers, Laboratoire Angevin de Recherche en Mathématiques（法国国家科学研究中心，昂热大学，安茹数学研究实验室）
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.
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,
