Sergei Yakovenko

On the number of zeros of Abelian integrals: A constructive solution of the infinitesimal Hilbert sixteenth problem

We prove that the number of limit cycles generated from nonsingular energy level ovals (periodic trajectories) in a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing infinitesimal Hilbert 16th problem.The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection defined over ℚ (the Gauss-Manin connection) with a quasiunipotent monodromy group.

Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

The tangential Hilbert 16th problem is to place an upper bound for the number of isolated ovals of algebraic level curves {H(x, y) = const} over which the integral of a polynomial 1-form P (x, y) dx + Q(x, y) dy (the Abelian integral) may vanish, the answer to be given in terms of the degrees n = deg H and d = max(deg P,deg Q). We describe an algorithm producing this upper bound in the form of a primitive recursive (in fact, elementary) function of n and d for the particular case of hyperelliptic polynomials H(x, y) = y2 + U(x) under the additional assumption that all critical values of U are real. This is the first general result on zeros of Abelian integrals that is completely constructive (i.e., contains no existential assertions of any kind). The paper is a research announcement preceding the forthcoming complete exposition. The main ingredients of the proof are explained and the differential algebraic generalization (that is the core result) is given. 1. Tangential Hilbert problem and bounds for the number of limit cycles in perturbed Hamiltonian systems 1.1. Complete Abelian integrals and the tangential Hilbert Sixteenth problem. Integrals of polynomial 1-forms over closed ovals of real algebraic curves, called (complete) Abelian integrals , naturally arise in many problems of geometry and analysis, but probably the most important is the link to the bifurcation of limit cycles of planar vector fields and the Hilbert Sixteenth problem. Recall that the question originally posed by Hilbert in 1900 was on the maximal number of limit cycles a polynomial vector field of degree d on the plane may have. This problem is still open even in the local version, for systems e-close to integrable or Hamiltonian ones. However, there is a certain hope that the “linearized”, or tangential Hilbert 16th problem can be more treatable. Consider a polynomial perturbation of a Hamiltonian polynomial vector field ẋ = − ∂y − eQ(x, y), ẏ = ∂H ∂x + eP (x, y). (1.1) An oval γ of the level curve H(x, y) = h which is a closed (but nonisolated) periodic trajectory for e = 0, may generate a limit cycle for small nonzero values of e only Received by the editors October 23, 1998. 1991 Mathematics Subject Classification. Primary 14K20, 34C05, 58F21; Secondary 34A20,

Generalized Rolle theorem in ℝ n and ℂ

AbstractThe Rolle theorem for functions of one real variable asserts that the number of zeros off on a real connected interval can be at most that off′ plus 1. The following inequality is a multidimensional generalization of the Rolle theorem: if ℓ[0,1] → ℝn,t→x(t), is a closed smooth spatial curve and L(ℓ) is the length of its spherical projection on a unit sphere, then for thederived curve ℓ′ [0,1], → ℝn

Complete Abelian integrals as rational envelopes

t \mapsto \dot x(t)

n

, the following inequality holds: L(ℓ) ⩽ L(ℓ′). For the analytic functionF(z) defined in a neighborhood of a closed plane curve Г ⊂ ℂ ≃ ℝ2 this inequality implies that

Bounded decomposition in the Brieskorn lattice and Pfaffian Picard–Fuchs systems for Abelian integrals☆

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