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Teruhisa TSUDA
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Hermite-Padé approximation, isomonodromic deformation and hypergeometric integral (with T. Mano)
Math. Z., in press.
We develop an underlying relationship between the theory of rational approximations and that of isomonodromic deformations. We show that a certain duality in Hermite's two approximation problems for functions leads to the Schlesinger transformations, i.e. transformations of a linear differential equation shifting its characteristic exponents by integers while keeping its monodromy invariant. Since approximants and remainders are described by block-Toeplitzs determinants, one can clearly understand the determinantal structure in isomonodromic deformations. We demonstrate our method in a certain family of Hamiltonian systems of isomonodromy type including the sixth Painlevé equation and Garnier systems; particularly, we present their solutions written in terms of iterated hypergeometric integrals. An algorithm for constructing the Schlesinger transformations is also discussed through vector continued fractions.


On a fundamental system of solutions of a certain hypergeometric equation
Ramanujan J. 38 (2015), 597--618
We study the linear Pfaffian systems satisfied by a certain class of hypergeometric functions, which includes Gauß's {}_2F_1, Thomae's {}_LF_{L-1} and Appell--Lauricella's F_D. In particular, we present a fundamental system of solutions with a characteristic local behavior by means of Euler-type integral representations. We also discuss how they are related to the theory of isomonodromic deformations or Painlevé equations.


Hypergeometric solution of a certain polynomial Hamiltonian system of isomonodromy type
Quart. J. Math. 63 (2012), 489--505
In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painlevé equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauß's hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.


UC hierarchy and monodromy preserving deformation
J. reine angew. Math. 690 (2014), 1--34
The UC hierarchy is an extension of the KP hierarchy, which possesses not only an infinite set of positive time evolutions but also that of negative ones. Through a similarity reduction we derive from the UC hierarchy a class of the Schlesinger systems including the Garnier system and the sixth Painlevé equation, which describes the monodromy preserving deformations of Fuchsian linear differential equations with certain spectral types. We also present a unified formulation of the above Schlesinger systems as a canonical Hamiltonian system whose Hamiltonian functions are polynomials in the canonical variables.


Projective reduction of the discrete Painlevé system of type A2 + A1 (with K. Kajiwara and N. Nakazono)
Int. Math. Res. Not. 2011 (2011), 930--966
We consider the q-Painlevé III equation arising from the birational representation of the affine Weyl group of type A2 + A1. We study the reduction of the q-Painlevé III equation to the q-Painlevé II equation from the viewpoint of affine Weyl group symmetry. In particular, the mechanism of apparent inconsistency between the hypergeometric solutions to both equations is clarified by using factorization of difference operators and the tau-functions.


From KP/UC hierarchies to Painlevé equations
Int. J. Math. 23 (2012), 1250010 (59pp)
We study the underlying relationship between Painlevé equations and infinite-dimensional integrable systems, such as the KP and UC hierarchies. We show that a certain reduction of these hierarchies by requiring homogeneity and periodicity yields Painlevé equations, including their higher order generalization. This result allows us to clearly understand various aspects of the equations, e.g., Lax formalism, Hirota bilinear relations for tau-functions, Weyl group symmetry, and algebraic solutions in terms of the character polynomials, i.e., the Schur function and the universal character.


Constructing two-dimensional integrable mappings that possess invariants of high degree (with J. Matsukidaira, A. Nobe and H. Tanaka)
RIMS Koukyuroku Bessatsu B13 (2009), 75--84
We propose a method for constructing two-dimensional integrable mappings that possess invariants with degree higher than two. Such integrable mappings are obtained by making a composition of a QRT mapping and a mapping that preserves the invariant curve of the QRT mapping except for changing the integration constant involved. We show several concrete examples whose integration constants change with period 2 and 3.


Ultradiscretization of a solvable two-dimensional chaotic map assciated with the Hesse cubic curve (with K. Kajiwara, M. Kaneko and A. Nobe)
Kyushu J. Math. 63 (2009), 315--338
We present a solvable two-dimensional piecewise linear chaotic map that arises from the duplication map of a certain tropical cubic curve. Its general solution is constructed by means of the ultradiscrete theta function. We show that the map is derived by the ultradiscretization of the duplication map associated with the Hesse cubic curve. We also show that it is possible to obtain the non-trivial ultradiscrete limit of the solution in spite of a problem known as 'the minus-sign problem.'


On an integrable system of q-difference equations satisfied by the universal characters: its Lax formalism and an application to q-Painlevé equations
Comm. Math. Phys. 293 (2010), 347--359
The universal character is a generalization of the Schur function attached to a pair of partitions. We study an integrable system of q-difference equations satisfied by the universal characters, which is an extension of the q-KP hierarchy and is called the lattice q-UC hierarchy. We describe the lattice q-UC hierarchy as a compatibility condition of its associated linear system (Lax formalism) and explore an application to the q-Painlevé equations via similarity reduction. In particular a higher-order analogue of the q-Painlevé VI equation is presented.


A geometric approach to tau-functions of difference Painlevé equations
Lett. Math. Phys. 85 (2008), 65--78
We present a unified description of birational representation of Weyl groups associated with T-shaped Dynkin diagrams, by using a particular configuration of points in the projective plane. A geometric formulation of tau-functions is given in terms of defining polynomials of certain curves. If the Dynkin diagram is of affine type (E_6, E_7 or E_8), our representation gives rise to the difference Painlevé equations.


Ultradiscretization of solvable one-dimensional chaotic maps (with K. Kajiwara and A. Nobe)
J. Phys. A: Math. Theor. 41 (2008), 395202 (13pp)
We consider the ultradiscretization of a solvable one-dimensional chaotic map which arises from the duplication formula of the elliptic functions. It is shown that ultradiscrete limit of the map and its solution yield the tent map and its solution simultaneously. A geometric interpretation of the dynamics of the tent map is given in terms of the tropical Jacobian of a certain tropical curve. Generalization to the maps corresponding to the m-th multiplication formula of the elliptic functions is also discussed.


A class of integrable and nonintegrable mappings and their dynamics (with B. Grammaticos, A. Ramani and T. Takenawa)
Lett. Math. Phys. 82 (2007), 39--49
We analyse a class of mappings which by construction do not belong to the QRT family. We show that some of the members of this class have invariants of high degree. A new linearisable mapping is also identified. A mapping which possesses confined singularities while having nonzero algebraic entropy is presented. Its dynamics are studied in detail and shown to be related intimately to the Fibonacci recurrence.


Tropical representation of Weyl groups associated with certain rational varieties (with T. Takenawa)
Adv. Math. 221 (2009), 936--954
Starting from certain rational varieties blown-up from (P^1)^N, we construct a tropical, i.e., subtraction-free birational, representation of Weyl groups as a group of pseudo isomorphisms of the varieties. We develop an algebro-geometric framework of tau-functions as defining functions of exceptional divisors on the varieties. In the case where the corresponding root system is of affine type, our construction yields a class of (higher order) q-difference Painlevé equations and its algebraic degree grows quadratically.


Tropical Weyl group action via point configurations and tau-functions of the q-Painlevé equations
Lett. Math. Phys. 77 (2006), 21--30
Starting from certain point sets in the projective plane, we construct a tropical (or subtraction-free birational) representation of Weyl groups over the field of tau-functions. In particular, our construction includes E_8, E_7, E_6 and D_5 as affine cases; each of them gives rise to the q-difference Painlevé equation.


Tau functions of q-Painlevé III and IV equations
Lett. Math. Phys. 75 (2006), 39--47
We present a geometric approach to tau-functions of the q-Painlevé III and IV equations via rational surfaces with affine Weyl group symmetry of type A_2+A_1. We also study a similarity reduction of the q-KP hierarchy to the equations.


Universal character and q-difference Painlevé equations
Math. Ann. 345 (2009), 395--415
The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we introduce an integrable system of q-difference lattice equations satisfied by the universal character, and call it the lattice q-UC hierarchy. We regard it as generalizing both q-KP and q-UC hierarchies. Suitable similarity and periodic reductions of the hierarchy yield the q-difference Painlevé equations of types A_{2g+1}, D_5, and E_6. As its consequence, a class of algebraic solutions of the q-Painlevé equations is rapidly obtained by means of the universal character. In particular, we demonstrate explicitly the reduction procedure for the case of type E_6, via the framework of tau-functions based on the geometry of certain rational surfaces.


q-Painlevé VI equation arising from q-UC hierarchy (with T. Masuda)
Comm. Math. Phys. 262 (2006), 595--609
We study the q-difference analogue of the sixth Painlevé equation (q-PVI) by means of tau functions associated with affine Weyl group of type D5. We prove that a solution of q-PVI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-PVI in terms of the universal character which is a generalization of Schur polynomial attached to a pair of partitions.


Universal characters and q-Painlevé systems
Comm. Math. Phys. 260 (2005), 59--73
We propose an integrable system of q-difference equations satisfied by the universal characters and regard it as a q-analogue of the UC hierarchy. Via a similarity reduction of this integrable system, rational solutions of the q-Painlevé systems are constructed in terms of the universal characters.


Universal characters, integrable chains and the Painlevé equations
Adv. Math. 197 (2005), 587--606
The universal character is a generalization of the Schur polynomial attached to a pair of partitions. We prove that the universal character solves the Darboux chain. The periodic closing of the chain is equivalent to the Painlevé equation of type A. Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters.


Tau functions of the fourth Painlevé equation in two variables
Funkcial. Ekvac. 48 (2005), 137--145
In this paper, we investigate the solutions of the fourth Painlevé equation in two variables by means of tau functions. Hirota bilinear forms and birational symmetries of the equation are presented. We obtain in particular a complete description of the rational solutions in terms of Schur polynomials.


Folding transformations of the Painlevé equations (with K. Okamoto and H. Sakai)
Math. Ann. 331 (2005), 713--738
New symmetries of the Painlevé differential equations, called folding transformations, are determined. These transformations are not birational but algebraic transformations of degree 2, 3 or 4. These are associated with quotients of the spaces of initial conditions of each Painlevé equation. We make the complete list of such transformations up to birational symmetries. We also discuss correspondences of special solutions of Painlevé equations.


Integrable mappings via rational elliptic surfaces
J. Phys. A: Math. Gen. 37 (2004), 2721--30
We present a geometric description of the QRT map (which is an integrable mapping introduced by Quispel, Roberts, and Thompson) in terms of the addition formula of a rational elliptic surface. By this formulation, we classify all the cases when the QRT map is periodic; and show that its period is 2, 3, 4, 5 or 6. A generalization of the QRT map which acts birationally on a pencil of K3 surfaces, or Calabi-Yau manifolds, is also presented.


Rational solutions of the Garnier system in terms of Schur polynomials
Int. Math. Res. Not. 2003 (2003), 2341--58
We present a family of rational solutions of the Garnier system; and show that the corresponding tau functions form special polynomials under a certain normalization. We give an explicit formula for the special polynomials in terms of the Schur polynomial attached to a rectangular Young diagram.


Universal characters and an extension of the KP hierarchy
Comm. Math. Phys. 248 (2004), 501--526
The universal character is a polynomial attached to a pair of partitions and is a generalization of the Schur polynomial. In this paper, we define vertex operators which play roles of raising operators for the universal character. By means of the vertex operators, we obtain a series of non-linear partial differential equations of infinite order, called the UC hierarchy; we regard it as an extension of the KP hierarchy.
We investigate also solutions of the UC hierarchy; we show that the totality of the space of solutions forms a direct product of two infinite-dimensional Grassmann manifolds, and describe its infinitesimal transformations in terms of infinite-dimensional Lie algebra.


Toda equation and special polynomials associated with the Garnier system
Adv. Math. 206 (2006), 657--683
We prove that a certain sequence of tau functions of the Garnier system satisfies Toda equation. We construct a class of algebraic solutions of the system by the use of Toda equation; then show that the associated tau functions are expressed in terms of the universal character, which is a generalization of Schur polynomial attached to a pair of partitions.


Birational symmetries, Hirota bilinear forms and special solutions of the Garnier systems in 2-variables
J. Math. Sci. Univ. Tokyo. 10 (2003), 355--371
Hirota bilinear forms of the Garnier system in 2-variables, G(1,1,1,1,1), are given. By using Hirota bilinear forms we construct new birational symmetries of G(1,1,1,1,1). We obtain special solutions of the Garnier system in n-variables, which are described in terms of solutions of the Garnier system in (n-1)-variables. We investigate also algebraic solutions for n=2.


Two approximation problems by Hermite, and Schlesinger transformations (in Japanese) (with T. Mano)
RIMS Kokyuroku Bessatsu B47 (2014), 77--86
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UC hierarchy, monodromy preserving deformation and hypergeometric function (in Japanese) ("Hook representation of the universal character" in Appendix C)
RIMS Kokyuroku 1765 (2011), 154--167
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UC hierarchy, monodromy preserving deformation and hypergeometric function (in Japanese)
Reports of RIAM Symposium 22AO-S8 (2011), 43--49
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Symmetrization of q-Painlevé equations (in Japanese) (with K. Kajiwara and N. Nakazono)
Reports of RIAM Symposium 20ME-S7 (2009), 21--28
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A geometric approach to tropical Weyl group actions and q-Painlevé equations (with T. Takenawa)
Oberwolfach Reports 5 (2008), 2588--2590
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Universal character and q-difference Painlevé equations (in Japanese)
RIMS Kokyuroku 1473 (2006), 1--14
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Infinite-dimensional integrable systems associated with universal characters, and Painlevé equations (in Japanese)
Reports of RIAM Symposium 16ME-S1 (2005), 128--136
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Integrable mappings via rational elliptic surfaces (in Japanese)
RIMS Kokyuroku 1400 (2004), 31--38
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Universal characters and an extension of the KP hierarchy (in Japanese)
RIMS Kokyuroku 1310 (2003), 54--64
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Toda equation and special polynomials associated with the Garnier system (in Japanese)
RIMS Kokyuroku 1296 (2002), 128--136
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Hierarchical structure and symmetries of the Garnier systems (in Japanese)
RIMS Kokyuroku 1203 (2001), 57--70
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Bilinear forms of the Garnier systems in two variables (in Japanese)
Rokko Lectures in Mathematics 7 (2000), 173--182
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Universal characters and Integrable systems
Ph.D. thesis, The University of Tokyo, 2003
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