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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.

Darboux developed an ingenious algebraic mechanism to construct infinite chains of ''integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this paper, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems ($\mathfrak{so}(3, C_K)$ systems) as well as a framework to extend Darboux transformations to any symmetric power of $\mathrm{SL}(2,\mathbb{C})$-systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.

Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group.

In this paper, we explain how to compute the Lie algebra of the differential Galois group of a reducible linear differential system. We achieve this by showing how to transform a block-triangular linear differential system into a Kolchin-Kovacic reduced form. We combine this with other reduction results to propose a general algorithm for computing a reduced form of a general linear differential system. In particular, this provides directly the Lie algebra of the differential Galois group without an a priori computation of this Galois group.

Consider a complex Hamiltonian system and an integral curve. In this paper, we give an effective and efficient procedure to put the variational equation of any order along the integral curve in reduced form provided that the previous one is in reduced form with an abelian Lie algebra. Thus, we obtain an effective way to check the Morales-Ramis-Sim\'o criterion for testing meromorphic Liouville integrability of Hamiltonian systems., Comment: 29 pages

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 Painlev\'e equations for special values of their parameter. In the Appendix, we recast the various notions of variational equations found in the literature and prove their equivalences., Comment: 35 pages. Keywords: Ordinary Differential Equations, Differential Galois Theory, Painlev\'e Equations, Computer Algebra

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.

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups. We show that these irreducible operators are not only globally nilpotent, but are such that they are homomorphic to their (formal) adjoints. Considering these operators, or, sometimes, equivalent operators, we show that they are also such that, either their symmetric square or their exterior square, have a rational solution. This is a general result: an irreducible linear differential operator homomorphic to its (formal) adjoint is necessarily such that either its symmetric square, or its exterior square has a rational solution, and this situation corresponds to the occurrence of a special differential Galois group. We thus define the notion of being "Special Geometry" for a linear differential operator if it is irreducible, globally nilpotent, and such that it is homomorphic to its (formal) adjoint. Since many Derived From Geometry n-fold integrals ("Periods") occurring in physics, are seen to be diagonals of rational functions, we address several examples of (minimal order) operators annihilating diagonals of rational functions, and remark that they also seem to be, systematically, associated with irreducible factors homomorphic to their adjoint., Comment: 62 pages

We present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field. Our approach is inspired by an idea of Ferragut and Giacomini. We improve upon their work by proving that rational first integrals can be computed via systems of linear equations instead of systems of quadratic equations. This leads to a probabilistic algorithm with arithmetic complexity $\bigOsoft(N^{2 \omega})$ and to a deterministic algorithm solving the problem in $\bigOsoft(d^2N^{2 \omega+1})$ arithmetic operations, where $N$ denotes the given bound for the degree of the rational first integral, and where $d \leq N$ is the degree of the vector field, and $\omega$ the exponent of linear algebra. We also provide a fast heuristic variant which computes a rational first integral, or fails, in $\bigOsoft(N^{\omega+2})$ arithmetic operations. By comparison, the best previous algorithm uses at least $d^{\omega+1}\, N^{4\omega +4}$ arithmetic operations. We then show how to apply a similar method to the computation of Darboux polynomials. The algorithms are implemented in a Maple package which is available to interested readers with examples showing its efficiency.

A differential system $[A] : \; Y'=AY$, with $A\in \mathrm{Mat}(n, \bar{k})$ is said to be in reduced form if $A\in \mathfrak{g}(\bar{k})$ where $\mathfrak{g}$ is the Lie algebra of the differential Galois group $G$ of $[A]$. In this article, we give a constructive criterion for a system to be in reduced form. When $G$ is reductive and unimodular, the system $[A]$ is in reduced form if and only if all of its invariants (rational solutions of appropriate symmetric powers) have constant coefficients (instead of rational functions). When $G$ is non-reductive, we give a similar characterization via the semi-invariants of $G$. In the reductive case, we propose a decision procedure for putting the system into reduced form which, in turn, gives a constructive proof of the classical Kolchin-Kovacic reduction theorem., Comment: To appear in : Journal of Pure and Applied Algebra

Let $\mathbf{k}$ be a differential field and let $[A]\,:\,Y'=A\,Y$ be a linear differential system where $A\in\mathrm{Mat}(n\,,\,\mathbf{k})$. We say that $A$ is in a reduced form if $A\in\mathfrak{g}(\bar{\mathbf{k}})$ where $\mathfrak{g}$ is the Lie algebra of $[A]$ and $\bar{\mathbf{k}}$ denotes the algebraic closure of $\mathbf{k}$. We owe the existence of such reduced forms to a result due to Kolchin and Kovacic \cite{Ko71a}. This paper is devoted to the study of reduced forms, of (higher order) variational equations along a particular solution of a complex analytical hamiltonian system $X$. Using a previous result \cite{ApWea}, we will assume that the first order variational equation has an abelian Lie algebra so that, at first order, there are no Galoisian obstructions to Liouville integrability. We give a strategy to (partially) reduce the variational equations at order $m+1$ if the variational equations at order $m$ are already in a reduced form and their Lie algebra is abelian. Our procedure stops when we meet obstructions to the meromorphic integrability of $X$. We make strong use both of the lower block triangular structure of the variational equations and of the notion of associated Lie algebra of a linear differential system (based on the works of Wei and Norman in \cite{WeNo63a}). Obstructions to integrability appear when at some step we obtain a non-trivial commutator between a diagonal element and a nilpotent (subdiagonal) element of the associated Lie algebra. We use our method coupled with a reasoning on polylogarithms to give a new and systematic proof of the non-integrability of the H\'enon-Heiles system. We conjecture that our method is not only a partial reduction procedure but a complete reduction algorithm. In the context of complex Hamiltonian systems, this would mean that our method would be an effective version of the Morales-Ramis-Sim\'o theorem., Comment: 15 pages

In this paper, we examine the non-relativistic stationary Schr\"odinger equation from a differential Galois-theoretic perspective. The main algorithmic tools are pullbacks of second order ordinary linear differential operators, so as to achieve rational function coefficients ("algebrization"), and Kovacic's algorithm for solving the resulting equations. In particular, we use this Galoisian approach to analyze Darboux transformations, Crum iterations and supersymmetric quantum mechanics. We obtain the ground states, eigenvalues, eigenfunctions, eigenstates and differential Galois groups of a large class of Schr\"odinger equations, e.g. those with exactly solvable and shape invariant potentials (the terms are defined within). Finally, we introduce a method for determining when exact solvability is possible., Comment: 62 pages

A theorem of N. Katz \cite{Ka} p.45, states that an irreducible differential operator $L$ over a suitable differential field $k$, which has an isotypical decomposition over the algebraic closure of $k$, is a tensor product $L=M\otimes_k N$ of an absolutely irreducible operator $M$ over $k$ and an irreducible operator $N$ over $k$ having a finite differential Galois group. Using the existence of the tensor decomposition $L=M\otimes N$, an algorithm is given in \cite{C-W}, which computes an absolutely irreducible factor $F$ of $L$ over a finite extension of $k$. Here, an algorithmic approach to finding $M$ and $N$ is given, based on the knowledge of $F$. This involves a subtle descent problem for differential operators which can be solved for explicit differential fields $k$ which are $C_1$-fields., Comment: 21 pages

A Swinging Atwood Machine (SAM) is built and some experimental results concerning its dynamic behaviour are presented. Experiments clearly show that pulleys play a role in the motion of the pendulum, since they can rotate and have non-negligible radii and masses. Equations of motion must therefore take into account the inertial momentum of the pulleys, as well as the winding of the rope around them. Their influence is compared to previous studies. A preliminary discussion of the role of dissipation is included. The theoretical behaviour of the system with pulleys is illustrated numerically, and the relevance of different parameters is highlighted. Finally, the integrability of the dynamic system is studied, the main result being that the Machine with pulleys is non-integrable. The status of the results on integrability of the pulley-less Machine is also recalled., Comment: 37 pages

Let $[A]: Y'=AY$ with $A\in \mathrm{M}_n (k)$ be a differential linear system. We say that a matrix $R\in {\cal M}_{n}(\bar{k})$ is a {\em reduced form} of $[A]$ if $R\in \mathfrak{g}(\bar{k})$ and there exists $P\in GL_n (\bar{k})$ such that $R=P^{-1}(AP-P')\in \mathfrak{g}(\bar{k})$. Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendants. In this article, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of hamiltonian systems. We use this to give an effective form of the Morales-Ramis theorem on (non)-integrability of Hamiltonian systems., Comment: 28 pages

We investigate a generalisation of the three dimensional spring-pendulum system. The problem depends on two real parameters $(k,a)$, where $k$ is the Young modulus of the spring and $a$ describes the nonlinearity of elastic forces. We show that this system is not integrable when $k\neq -a$. We carefully investigated the case $k= -a$ when the necessary condition for integrability given by the Morales-Ramis theory is satisfied. We discuss an application of the higher order variational equations for proving the non-integrability in this case., Comment: 20 pages, 1 figure