Emil Horozov

Linear estimate for the number of zeros of Abelian integrals with cubic Hamiltonians

An explicit upper bound is derived for the number of the zeros of the integral of degree n polynomials f, g, on the open interval for which the cubic curve contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals , i,j=0,1, generating the module of complete Abelian integrals I(h) (over the ring of polynomials in h).

General methods for constructing bispectral operators

We present methods for obtaining new solutions to the bispectral problem. We achieve this by giving its abstract algebraic version suitable for generalizations. All methods are illustrated by new classes of bispectral operators.

Bispectral Algebras of Commuting Ordinary Differential Operators

Abstract:We develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank N. It combines and unifies the ideas of Duistermaat–Grünbaum and Wilson. Our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators. The method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem.

The Pfaff lattice and skew-orthogonal polynomials

Consider a semi-inflnite skew-symmetric moment matrix, m1 evolving according to the vector flelds @m=@tk = ⁄ k m + m⁄ >k ; where ⁄ is the shift matrix. Then The skew-Borel decomposition m1 := Q i1 JQ >i1 leads to the so-called Pfafi Lattice, which is integrable, by virtue of the AKS theorem, for a splitting involving the a‐ne symplectic algebra. The tau-functions for the system are shown to be pfa‐ans and the wave vectors skew-orthogonal polynomials; we give their explicit form in terms of moments. This system plays an important role in symmetric and symplectic matrix models and in the theory of random matrices (beta=1 or 4).

The solution to the q-KdV equation

Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this Letter is to show that any KdV solution leads effectively to a solution of the q-approximation of KdV. Two different q-KdV approximations were proposed, first one by Frenkel [Int. Math. Res. Notices 2 (1996) 55] and a variation by Khesin, Lyubashenko and Roger [J. Func. Anal. 143 (1997) 55]. We show there is a dictionary between the solutions of q-KP and the 1-Toda lattice equations, obeying some special requirement; this is based on an algebra isomorphism between difference operators and D-operators, where Df(x) = f(qx). Therefore every notion about the 1-Toda lattice can be transcribed into q-language, (C) 1998 Elsevier Science B.V.

Highest weight modules over the

The present paper establishes a connection between the Lie algebra W_{1+infty} and the bispectral problem. We show that the manifolds of bispectral operators obtained by Darboux transformations on powers of Bessel operators are in one to one correspondence with the manifolds of tau-functions lying in the W_{1+infty}-modules M_beta introduced in our previous paper hep-th/9510211. An immediate corollary is that they are preserved by hierarchies of symmetries generated by subalgebras of W_{1+infty}. This paper is the last of a series of papers (hep-th/9510211, q-alg/9602010, q-alg/9602011) on the bispectral problem.

W_{1+\infty}

The paper studies the Painlevé VIe equations from the point of view of Hamiltonian nonintegrability. For certain infinite number of points in the parameter space we prove that the equations are not integrable. Our approach uses recent advance in Hamiltonian integrability reducing the problem to higher differential Galois groups as well as the monodromy of dilogarithic functions.

algebra and the bispectral problem

For each r = (r_1, r_2,...,r_N) we construct a highest weight module M_r of the Lie algebra W_{1+infty}. The highest weight vectors are specific tau-functions of the N-th Gelfand--Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors. This paper is the first of a series of papers (q-alg/9602010, q-alg/9602011, q-alg/9602012) on the bispectral problem.For each we construct a highest weight module of the Lie algebra The highest weight vectors are specific tau-functions of the Nth Gelfand - Dickey hierarchy. We show that these modules are quasifinite and we give a complete description of the reducible ones together with a formula for the singular vectors.

Non-integrability of some Painlevé VI-equations and dilogarithms

It is proved that arbitrary quadratic perturbations of quadratic Hamiltonian systems in the plane possessing central symmetry can produce at most two limit cycles.

Tau-functions as highest weight vectors for algebra

Abstract For the system describing the motion of the spherical pendulum we find explicitly the set where the condition of isoenergetical non-degeneracy important for KAM-theory is violated. The result is interpreted also in terms of the behaviour of the increment of the azimuth when the angular momentum varies.