Differential Galois Group Research Articles

Galois Correspondence Theorem for Picard-Vessiot Extensions

For a homogeneous linear differential equation defined over a differential field K, a Picard-Vessiot extension is a differential field extension of K differentially generated by a fundamental system of solutions of the equation and not adding constants. When K has characteristic 0 and the field of constants of K is algebraically closed, it is well known that a Picard-Vessiot extension exists and is unique up to K-differential isomorphism. In this case the differential Galois group is defined as the group of K-differential automorphisms of the Picard-Vessiot extension and a Galois correspondence theorem is settled. Recently, Crespo, Hajto and van der Put have proved the existence and unicity of the Picard-Vessiot extension for formally real (resp. formally p-adic) differential fields with a real closed (resp. p-adically closed) field of constants. This result widens the scope of application of Picard-Vessiot theory beyond the complex field. It is then necessary to give an accessible presentation of Picard-Vessiot theory for arbitrary differential fields of characteristic zero which eases its use in physical or arithmetic problems. In this paper, we give such a presentation avoiding both the notions of differential universal extension and specializations used by Kolchin and the theories of schemes and Hopf algebras used by other authors. More precisely, we give an adequate definition of the differential Galois group as a linear algebraic group and a new proof of the Galois correspondence theorem for a Picard-Vessiot extension of a differential field with non algebraically closed field of constants, which is more elementary than the existing ones.

Non-integrability of restricted double pendula

We consider two special types of double pendula, with the motion of masses restricted to various surfaces. In order to get quick insight into the dynamics of the considered systems the Poincaré cross sections as well as bifurcation diagrams have been used. The numerical computations show that both models are chaotic which suggests that they are not integrable. We give an analytic proof of this fact checking the properties of the differential Galois group of the system's variational equations along a particular non-equilibrium solution.

Integrability of Hamiltonian systems with algebraic potentials

Problem of integrability for Hamiltonian systems with potentials that are algebraic thus multivalued functions of coordinates is discussed. Introducing potential as a new variable the original Hamiltonian system on 2n dimensional phase space is extended to 2n+1 dimensional system with rational right-hand sides. For extended system its non-canonical degenerated Poisson structure of constant rank 2n and rational Hamiltonian is identified. For algebraic homogeneous potentials of non-zero rational homogeneity degree necessary integrability conditions are formulated. These conditions are deduced from an analysis of the differential Galois group of variational equations around particular solutions of a straight line type. Obtained integrability obstructions are applied to the class of monomial homogeneous potentials. Some integrable potentials satisfying these conditions are found.

Note on integrability of certain homogeneous Hamiltonian systems

In this paper we investigate a class of natural Hamiltonian systems with two degrees of freedom. The kinetic energy depends on coordinates but the system is homogeneous. Thanks to this property it admits, in a general case, a particular solution. Using this solution we derive necessary conditions for the integrability of such systems investigating differential Galois group of variational equations.

Non-Integrability of Some Higher-Order Painlevé Equations in the Sense of Liouville

In this paper we study the equation $$ w^{(4)} = 5 w" (w^2 - w') + 5 w (w')^2 - w^5 + (\lambda z + \alpha)w + \gamma, $$ which is one of the higher-order Painlev\'e equations (i.e., equations in the polynomial class having the Painlev\'e property). Like the classical Painlev\'e equations, this equation admits a Hamiltonian formulation, B\"acklund transformations and families of rational and special functions. We prove that this equation considered as a Hamiltonian system with parameters $\gamma/\lambda = 3 k$, $\gamma/\lambda = 3 k - 1$, $k \in \mathbb{Z}$, is not integrable in Liouville sense by means of rational first integrals. To do that we use the Ziglin-Morales-Ruiz-Ramis approach. Then we study the integrability of the second and third members of the $\mathrm{P}_{\mathrm{II}}$-hierarchy. Again as in the previous case it turns out that the normal variational equations are particular cases of the generalized confluent hypergeometric equations whose differential Galois groups are non-commutative and hence, they are obstructions to integrability.

Computing differential Galois groups of parameterized second-order linear differential equations

Computing differential Galois groups of parameterized second-order linear differential equations

An exactly solvable system from quantum optics

We investigate a generalisation of the Rabi system in the Bargmann–Fock representation. In this representation the eigenproblem of the considered quantum model is described by a system of two linear differential equations with one independent variable. The system has only one irregular singular point at infinity. We show how the quantisation of the model is related to asymptotic behaviour of solutions in a vicinity of this point. The explicit formulae for the spectrum and eigenfunctions of the model follow from an analysis of the Stokes phenomenon. An interpretation of the obtained results in terms of differential Galois group of the system is also given.

Real Liouville Extensions

We characterize linear differential equations defined over a real differential field with a real closed field of constants C, which are solvable by real Liouville functions, as those having a differential Galois group whose identity component is solvable and C-split.

Scaling functions in the square Ising model

We show and give the linear differential operators of order , for the integrals which appear in the two-point correlation scaling function of Ising model . The integrals are given in expansion around in the basis of the formal solutions of with transcendental combination coefficients. We find that the expression is a solution of the Painlevé VI equation in the scaling limit. Combinations of the (analytic at ) solutions of sum to . We show that the expression is the scaling limit of the correlation function and . The differential Galois groups of the factors occurring in the operators are given.

Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups

We first revisit an order-six linear differential operator, already introduced in a previous paper, having a solution which is a diagonal of a rational function of three variables. This linear differential operator is such that its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We generalize this decomposition to decomposition in terms of three self-adjoint operators of arbitrary orders, provided the three orders have the same parity. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators, with selected differential Galois groups, is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi–Yau three-folds, obtained recently by Lairez, correspond to a particular form of the decomposition detailed in this paper. All the results of this paper can be seen as providing an algebraic characterization of linear differential operators with selected symplectic or orthogonal differential Galois groups.

On the parameterized differential inverse Galois problem over [formula omitted

In this article, we consider the inverse Galois problem for parameterized differential equations over k((t))(x) with k any field of characteristic zero and use the method of patching over fields due to Harbater and Hartmann. We show that every connected semisimple k((t))-split linear algebraic group is a parameterized differential Galois group over k((t))(x).

Lattice Green functions: the seven-dimensional face-centred cubic lattice

We present a recursive method to generate the expansion of the lattice Green function of the d-dimensional face-centred cubic (fcc) lattice. We produce a long series for d = 7. Then we show (and recall) that, in order to obtain the linear differential equation annihilating such a long power series, the most economic way amounts to producing the non-minimal order differential equations. We use the method to obtain the minimal order linear differential equation of the lattice Green function of the seven-dimensional fcc lattice. We give some properties of this irreducible order-eleven differential equation. We show that the differential Galois group of the corresponding operator is included in . This order-eleven operator is non-trivially homomorphic to its adjoint, and we give a ‘decomposition’ of this order-eleven operator in terms of four order-one self-adjoint operators and one order-seven self-adjoint operator. Furthermore, using the Landau conditions on the integral, we forward the regular singularities of the differential equation of the d-dimensional lattice and show that they are all rational numbers. We evaluate the return probability in random walks in the seven-dimensional fcc lattice. We show that the return probability in the d-dimensional fcc lattice decreases as d−2 as the dimension d goes to infinity.

Lax operators, Poisson groups, and the differential Galois theory

We examine the transfer of a Poisson structure in a space of differential or difference Lax operators to the space of solutions of the corresponding auxiliary problem (the space of wave functions) in detail. We investigate a spontaneous symmetry breaking resulting in the appearance of a nontrivial Poisson structure on the differential or difference Galois group. We review the difference version of the Drinfeld-Sokolov theory and describe a new type of classical r-matrices related to generalized exchange algebras.

A density theorem in parametrized differential Galois theory

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 also a necessary condition on a group for being a global parameterized differential Galois group; as a corollary of the density theorem, we prove that their condition is also sufficient. As an application, we give a characterization of completely integrable equations, and we give a partial answer to a question of Sibuya about the transcendence properties of a given Stokes matrix. Moreover, using a parameterized Hukuhara-Turrittin theorem, we show that the Galois group descends to a smaller field, whose field of constants is not differentially closed.

The Ising model and special geometries

We show that the globally nilpotent G-operators corresponding to the factors of the linear differential operators annihilating the multifold integrals χ(n) of the magnetic susceptibility of the Ising model (n ⩽ 6) are homomorphic to their adjoint. This property of being self-adjoint up to operator homomorphisms is equivalent to the feature of their symmetric squares, or their exterior squares, having rational solutions. The differential Galois groups are in the special orthogonal, or symplectic, groups. This self-adjoint (up to operator equivalence) property means that the factor operators that we already know to be derived from geometry are special globally nilpotent operators: they correspond to ‘special geometries’. Beyond the small order factor operators (occurring in the linear differential operators associated with χ(5) and χ(6)), and, in particular, those associated with modular forms, we focus on the quite large order-12 and order-23 operators. We show that the order-12 operator has an exterior square which annihilates a rational solution. Then, its differential Galois group is in the symplectic group . The order-23 operator is shown to factorize into an order-2 operator and an order-21 operator. The symmetric square of this order-21 operator has a rational solution. Its differential Galois group is, thus, in the orthogonal group .

Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations

The present work is the first one of two papers, in which we analyse systems of higher order variational equations associated to natural Hamiltonian systems with homogeneous potential of degree $k\in\mathbb{Z}\setminus \{-1,0,1\}$. Our attempt is to give necessary conditions for complete integrability which can be deduced in a framework of differential Galois theory. We show that the higher variational equations $\mathrm{VE}_p$ of order $p\geq 2$, although complicated, have a very particular algebraic structure. More precisely, we show that if $\mathrm{VE}_1$ has virtually Abelian differential Galois group (DGG), then $\mathrm{VE}_{p}$ are solvable for an arbitrary $p>1$. We proved this inductively using what we call the second level integrals. Then we formulate the necessary and sufficient conditions in terms of these second level integrals for $\mathrm{VE}_{p}$ to be virtually Abelian. We apply the above conditions to potentials of degree $k=\pm 2$ considering their $\mathrm{VE}_p$ with $p>1$ along Darboux points. For $k= 2$, $\mathrm{VE}_1$ does not give any obstruction to the integrability. We show that under certain non-resonance condition, the only degree two integrable potential is the multidimensional harmonic oscillator. In contrast, for degree $k=-2$ potentials, all the $\mathrm{VE}_{p}$ along Darboux points are virtually Abelian.

Non-integrability of the fourth Painlevé equation in the Liouville–Arnold sense

In this paper we are concerned with the integrability of the fourth Painlevé equation (PIV) from the point of view of the Hamiltonian dynamics. We prove that the fourth Painlevé equation0.1with parameters a = m, b = −2(1 + 2n + m) where , is not integrable in the Liouville–Arnold sense by means of meromorphic first integrals. We explicitly compute formal and analytic invariants of the second variational equations which generate topologically the differential Galois group. In this way our calculations and the Ziglin–Ramis–Morales-Ruiz–Simó method yield the non-integrability results.

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 Ru(H) of the differential Galois group H associated to a parameterized second-order homogeneous linear differential equation of the form∂2∂x2Y−qY=0, where q∈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 Ru(H) to solving a creative telescoping problem, whose effective solution depends on the assumption that the maximal reductive quotient H/Ru(H) is a Π-constant linear differential algebraic group. When this condition is not satisfied, one can effectively compute a new set of parametric derivations Π′ such that the associated differential Galois group H′ has the property that H′/Ru(H′) is Π′-constant, and such that Ru(H) is defined by the same differential equations as Ru(H′). Thus the computation of Ru(H) is reduced to the effective computation of Ru(H′). We expect that an elaboration of this method will be successful in extending the applicability of some recent algorithms developed by Minchenko, Ovchinnikov, and Singer to compute unipotent radicals for higher order equations.

On linear differential equations with reductive Galois group

Given a connection on a meromorphic vector bundle over a compact Riemann surface with reductive differential Galois group, we associate to it a projective variety. Connections such that their associated projective variety is curves can be classified, up to projective equivalence, using ruled surfaces. In particular, such a meromorphic connection is the pullback of a standard connection. This extends a similar result by Klein for second-order ordinary linear differential equations to a broader class of equations.

Differential algebra on lattice Green functions and Calabi–Yau operators

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi–Yau operators and order-7 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.