Multiple zeta values and the WKB method

The multiple zeta values ζ(d1, . . . , dr ) are natural generalizations of the values ζ(d) of the Riemann zeta functions at integers d. They have many applications, e.g. in knot theory and in quantum physics. It turns out that some generating functions for the multiple zeta values, like fd(x) = 1 − ζ(d)xd + ζ(d, d)x2d − . . . , are related with hypergeometric equations. More precisely, fd(x) is the value at t = 1 of some hypergeometric series dFd−1(t) = 1 − x t + . . ., a solution to a hypergeometric equation of degree d with parameter x. Our idea is to represent fd(x) as some connection coeffi- cient between certain standard bases of solutions near t = 0 and near t = 1. Moreover, we assume that |x| is large. For large complex x the above basic solutions are represented in terms of so-called WKB solutions. The series which define the WKB solutions are divergent and are subject to so-called Stokes phenomenon. Anyway it is possible to treat them rigorously. In the paper we review our results about application of the WKB method to the generating functions f x), focusing on the cases d = 2 and d = 3

Small amplitude limit cycles for the polynomial Liénard system

AbstractWe estimate for the maximal number of limit cycles bifurcating from a focus for the Liénard equation x¨+f(x)x˙+g(x)=0, where f and g are polynomials of degree m and n respectively. These estimates are quadratic in m and n and improve the existing bounds. In the proof we use methods of complex algebraic geometry to bound the number of double points of a rational affine curve

The restricted three body problem revisited

AbstractWe present a new computation of the Birkhoff normal form for the Hamiltonian of the restricted three body problem near the Lagrangian libration points. This leads to a new proof of the Lyapunov stability of these points

The Analytic and Formal Normal Form for the Nilpotent Singularity

AbstractWe study orbital normal forms for analytic planar vector fields with nilpotent singularity. We show that the Takens normal form is analytic. In the case of generalized cusp we present the complete formal orbital normal form; it contains functional moduli. We interprete the coefficients of these moduli in terms of the hidden holonomy group

Note on the deck transformations group and the monodromy group

For a ramified covering between Riemann surfaces the groups Deck of deck transformations and Mon of monodromy permutations are introduced. We associate with them groups of automorphisms of certain extensions of function fields. We study relations between these objects. \endabstrac

On first integrals for polynomial differential equations on the line

We show that any equation dy/dx = P(x,y) with P a polynomial has a global (on ℝ²) smooth first integral nonconstant on any open domain. We also present an example of an equation without an analytic primitive first integral

The XVI-th Hilbert problem about limit cycles

1. Introduction. The XVI-th Hilbert problem consists of two parts. The first part concerns the real algebraic geometry and asks about the topological properties of real algebraic curves and surfaces. The second part deals with polynomial planar vector fields and asks for the number and position of limit cycles. The progress in the solution of the first part of the problem is significant. The classification of algebraic curves in the projective plane was solved for degrees less than 8. Among general results we notice the inequalities of Harnack and Petrovski and Rohlin's theorem. Other results were obtained by Newton, Klein, Clebsch, Hilbert, Nikulin, Kharlamov, Gudkov, Arnold, Viro, Fidler. There are multidimensional generalizations: the theory of Khovansky, the inequalities of Petrovski and Oleinik, and others. In contrast to the algebraic part of the problem, the progress in the solution of the second part is small. In the present article we concentrate on the second part of the XVI-th Hilbert problem

The topological proof of Abel-Ruffini theorem

We present a proof of the non-solvability in radicals of a general algebraic equation of degree greater than four. This proof relies on the non-solvability of the monodromy group of a general algebraic function

On algebraic solutions of algebraic Pfaff equations

We give a new proof of Jouanolou's theorem about non-existence of algebraic solutions to the system $ẋ=z^{s}, ẏ=x^{s}, ż=y^{s}$. We also present some generalizations of the results of Darboux and Jouanolou about algebraic Pfaff forms with algebraic solutions