We prove that in quadratic perturbations of generic quadratic Hamiltonian vector fields with three saddle points and one centre there can appear at most two limit cycles. This bound is exact.

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.

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.q-alg/9605011 IntroductionThe bispectral problem of J. J. Duistermaat and F. A. Grünbaum [9] consists of finding all bispectral ordinary differential operators, i.e. operators L(x, ∂ x ) having a family of eigenfunctions ψ(x, z), which are also eigenfunctions for another differential operator Λ(z, ∂ z ) in the spectral parameter:In [3] we constructed large families of solutions to this problem, generalizing all previously known results (cf. [9,19,16,17,13], etc.). We obtained them as special Darboux transformations (called "polynomial") from the most obvious solutionsthe (generalized) Bessel and Airy ones. We recall that the Bessel (respectivelyAiry) operators have the formFor fixed ψ, all operators L(x, ∂ x ) for which ψ is an eigenfunction form a commutative algebra A ψ , called the spectral algebra. Then SpecA ψ is an algebraic curve [6] -the spectral curve. The dimension of the space of eigenfunctions ψ (= g.c.d. ordL, L ∈ A ψ ) is called a rank of A ψ . Following G. Wilson [17] we call the spectral algebra A ψ bispectral iff there exists an operator Λ(z, ∂ z ) satisfying (0.2).The purpose of this paper is to present general methods for constructing bispectral operators (respectively -algebras). Some of our statements are only abstract *

An explicit upper bound Z(3, n) 5n+15 is derived for the number of the zeros of the integral h → I (h) = H =h g(x, y) dx −f (x, y) dy of degree n polynomials f, g, on the open interval for which the cubic curve {H = h} contains an oval. The proof exploits the properties of the Picard-Fuchs system satisfied by the four basic integrals H

Let KdV stand for the Nth Gelfand-Dickey reduction of the KP hierarchy. The purpose of this paper 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 E. Frenkel [7] and a variation by Khesin, Lyubashenko and Roger [10]. 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.Consider the q-difference operators D and D q , defined by Df (y) = f (qy) and D q f (y) := f (qy) − f (y) (q − 1)y , and the q-pseudo-differential operators

No abstract

This paper is the last of a series of papers devoted to the bispectral problem [3]- [6].Here we examine the connection between the bispectral operators constructed in [6] and the Lie algebra W 1+∞ (and its subalgebras). To give a more detailed idea of the contents of the present paper we briefly recall the results of [4]-[6] which we need.In [4] we built large families of representations of W 1+∞ . For each β ∈ C N we defined a tau-function τ β (t) which we called Bessel tau-function. We proved that it is a highest weight vector for a representation M β of the algebra W 1+∞ with central charge N . In [6] we introduced a version of Darboux transformation, which we called monomial, on the corresponding wave functions Ψ β (x, z) (see also Subsect. 1.2) and showed that the resulting wave functions are bispectral. For example all bispectral operators from [9,22] can be obtained in this way.The present paper establishes closer connections between W 1+∞ and the bispectral problem. Our first result (Theorem 2.1) shows that a tau-function is a monomial Darboux transformation of a Bessel tau-function if and only if it belongs to one of the modules M β . This type of connection between the representation theory (of W 1+∞ ) and the bispectral problem is, to the best of our knowledge, new even for the bispectral tau-functions of Duistermaat and Grünbaum [9].The second of the questions we try to answer in the present paper originates from Duistermaat and Grünbaum [9]. They noticed that their rank 1 bispectral operators are invariant under the KdV-flows and asked if there is a hierarchy of symmetries for the rank 2 bispectral operators. The latter question was answered affirmatively by Magri and Zubelli [17] who showed that the algebra V ir + (the subalgebra of the Virasoro algebra spanned by the operators of non-negative weight) is tangent to *

In recent years, the Kowalewski top has received a lot of attention, not only because of the ingenious way Sophie Kowalewskaya integrates her system, but also because of its hidden symmetries. Exactly one hundred years ago, Kowalewski [14], [15] found that not only are the Euler and Lagrange rigid body motions integrable in terms of Abelian integrals, but also a third solid body motion-named after her. In her famous Acta Mathematica paper, she integrates the problem in terms of hyperelliptic integrals, using a very beautiful but mysterious change of variables. When some of us looked at that problem we amved at what seemed like a different conclusion. Namely, we found that the invariant surfaces could be completed, via the flow, into complex algebraic tori (Abelian surfaces) T 2 = Cz/A, where the lattice A is spanned by the columns of the period matrix. The paradox alluded to above is well known in algebraic geometry. On the one hand taking half the second period in (1) turns the Abelian surface T 2 into a new Abelian surface J of which TZ is a double covering, as discussed in Section 5. J is principally polarized, as the jargon calls it, and is thus the Jacobian of a 'In fact, W : Bath has pointed out that Abelian surfaces with period matrix (11, which are embeddable into P', are easier to study than the customary principally polarized surfaces, which can at best be embedded into Ps.