Fuchsian Systems Research Articles

Additive Deligne-Simpson Problem for Non-Fuchsian Systems

We consider the problem for which tuples of conjugacy classes cj ⊂ gl (n, C) and for which tuples of non-resonant formal normal forms does there exist a non-Fuchsian linear system of ordinary differential equations on Riemann's sphere with residue matrices from the classes cj and with the given formal normal forms at its poles. It turns out that the answer is the same as in the case of Fuchsian systems and depends only on the conjugacy classes cj. We find also the dimension of the moduli space of such non-Fuchsian systems.

Topology of complements of hyperplane arrangements and isomonodromic deformations of fuchsian systems

We consider families of systems of first-order linear differential equations on complex linear spaces that represent the Cherednik variant of the Knizhnik-Zamolodchikov equations. For these families of equations, we prove a rigidity property with respect to a certain class of isomonodromic deformations; i.e., we show the absence of nontrivial special isomonodromic deformations with a movable divisor of singularities.

Fuchsian systems with completely reducible monodromy

The solvability of the Riemann-Hilbert problem for representations χ = χ 1 ⊕ χ 2 having the form of a direct sum is considered. It is proved that any representation χ 1 can be realized as a direct summand in the monodromy representation χ of a Fuchsian system. Other results are also obtained, which suggest a simple method for constructing counterexamples to the Riemann-Hilbert problem.

A hypergeometric system of the Heun equation and middle convolution

A new method of obtaining integral transformation of the Heun equation is described. The Heun equation is reduced to the so-called hypergeometric system, a linear Fuchsian system of rank 3, and a Detweiler and Reiter algebraic analogue of the Katz middle convolution functor is applied. A linear Fuchsian system of rank 2 associated with the Heun equation is also studied.

Middle Convolution and Heun's Equation

Heun's equation naturally appears as special cases of Fuchsian system of differential equations of rank two with four singularities by introducing the space of ini- tial conditions of the sixth Painleve equation. Middle convolutions of the Fuchsian system are related with an integral transformation of Heun's equation.

Polynomial bounds for the oscillation of solutions of Fuchsian systems

Nous etudions le probleme d'une borne superieure effective sur le nombre des racines isolees des solutions de systemes de type Fuchs sur la sphere de Riemann. Le resultat principal est une borne explicite non uniforme a croissance polynomiale sur la frontiere de l'ensemble des systemes fuchsiens de dimension n quelconque ayant m singularites. Comme une fonction de n, m, la borne est doublement exponentielle dans le sens precis decrit dans le manuscrit. Comme corollaire, nous obtenons la solution a croissance polynomiale du probleme d'Hilbert infinitesimal restreint, qui ameliore les bornes exponentielles recemment obtenues par A. Glutsyuk et Yu. Ilyashenko.

Analytic linearization of nonlinear perturbations of Fuchsian systems

Nonlinear perturbations of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable.

Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms

Nonlinear perturbations of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a unique formal ‘correction’ of the nonlinear part so that the ‘corrected’ system is formally linearizable.Normal forms of these systems are found, providing also their classification.

Analytic methods in quantum computing

Here we consider a model of quantum computation, based on the monodromy representation of a Fuchsian system. The rôle of local and entangling operators in monodromic quantum computing is played by monodromy matrices of connections with logarithmic singularities acting on the fiber of a holomorphic vector bundle as on the space of qubits. The leading theme is the problem of constructing a set of universal gates as monodromy operators induced from a connection with logarithmic singularity. In the formal scheme developed by us, already known models — topological and holonomic — can be incorporated.

Euler integral symmetries for a deformed Heun equation and symmetries of the Painlevé PVI equation

Euler integral transformations relate solutions of ordinary linear differential equations and generate integral representations of the solutions in a number of cases or relations between solutions of constrained equations (Euler symmetries) in some other cases. These relations lead to the corresponding symmetries of the monodromy matrices. We discuss Euler symmetries in the case of the simplest Fuchsian system that is equivalent to a deformed Heun equation, which is in turn related to the Painleve PVI equation. The existence of integral symmetries of the deformed Heun equation leads to the corresponding symmetries of the PVI equation.

On the Painlevé Property of Isomonodromic Deformations of Fuchsian Systems

We give a review of the modern theory of isomonodromic deformations of Fuchsian systems discussing both classical and modern results, such as a general form of the isomonodromic deformations of Fuchsian systems, their differences from the classical Schlesinger deformations, the Fuchsian system moduli space structure and the geometric meaning of new degrees of freedom appeared in a non-Schlesinger case. Using this we illustrate some general relations between such concepts as integrability, isomonodromy and Painleve property.

Familles fuchsiennes d’équations aux ($q$-)différences et confluence

In a first part, we give a method for solving a family of fuchsian systems of operators of pseudo-derivations associated to a family of homographies with two parameters which unify and generalize the differential, the difference and the $q$-difference cases. In a second part, we study the problems of confluence related to these families.

Integral representation of solutions to Fuchsian system and Heun's equation

We obtain integral representations of solutions to special cases of the Fuchsian system of differential equations and Heun's differential equation. In particular, we calculate the monodromy of solutions to the Fuchsian equation that corresponds to Picard's solution of the sixth Painlevé equation, and to Heun's equation.

Middle convolution of Fuchsian systems and the construction of rigid differential systems

In [M. Dettweiler, S. Reiter, An algorithm of Katz and its application to the inverse Galois problem, J. Symbolic Comput. 30 (2000) 761–798], a purely algebraic analogon of Katz' middle convolution functor (see [N.M. Katz, Rigid Local Systems, Ann. of Math. Stud., vol. 139, Princeton University Press, 1997]) is given. In this paper, we find an explicit Riemann–Hilbert correspondence for this functor. This leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the Riemann–Hilbert correspondence.

Middle convolution and deformation for Fuchsian systems

Middle convolution and addition are operations for Fuchsian systems of differential equations which preserve the number of accessory parameters. In this paper we show that they also preserve the deformation equations. Several Bäcklund transformations of the sixth Painlevé equation are obtained from this viewpoint.

Painlevé equations and the middle convolution

Abstract We relate the theory of the middle convolution functor MCλ to the study of algebraic solutions of the sixth Painlevé equation PVI. We interpret some recently found algebraic solutions by Boalch in terms of the middle convolution. Then we show how to obtain algebraic solutions of PVI by starting with triples in GL2 and applying MCλ . The effect on the underlying Fuchsian systems can be described in a very simple manner.

General linear problem of the isomonodromic deformation of Fuchsian systems

In contrast to nonresonance systems whose continuous deformations are always Schlesinger deformations, systems with resonances provide great possibilities for deformations. In this case, the number of continuous parameters of deformation, in addition to the location of the poles of the system, includes the data describing the Levelt structure of the system, or, in other words, the distribution of resonance directions in the space of solutions. The question of classifying the form and structure of deformations according to these parameters arises. In the present paper, we consider continuous isomonodromic deformations of Fuchsian systems, including those with respect to additional parameters, describe the corresponding linear problem, and present the Pfaff form of the linear problem of general continuous isomonodromic deformation of Fuchsian systems.

Monodromy of Fuchsian systems on complex linear spaces

On complex linear spaces, Fuchs-type Pfaffian systems are studied that are defined by configurations of vectors in these spaces. These systems are referred to as R-systems in this paper. For the vector configurations that are systems of roots of complex reflection groups, the monodromy representations of R-systems are described. These representations are deformations of the standard representations of reflection groups. Such deformations define representations of generalized braid groups corresponding to complex reflection groups and are similar to the Burau representations of the Artin braid groups.

Canonical Structure and Symmetries of the Schlesinger Equations

The Schlesinger equations S (n,m) describe monodromy preserving deformations of order m Fuchsian systems with n + 1 poles. They can be considered as a family of commuting time-dependent Hamiltonian systems on the direct product of n copies of m × m matrix algebras equipped with the standard linear Poisson bracket. In this paper we present a new canonical Hamiltonian formulation of the general Schlesinger equations S (n,m) for all n, m and we compute the action of the symmetries of the Schlesinger equations in these coordinates.

Irreducible Fuchsian system with reducible monodromy representation

We present an example of the reducible representation χ = χ1 ⨁ χ2, which, on the one hand, is the monodromy representation of a Fuchsian system. On the other hand, the representation χ2 is a counterexample to the Riemann-Hilbert problem. Using a meromorphic gauge transformation, one cannot reduce this system to the direct sum of Fuchsian systems corresponding to the subrepresentations.