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 = degH and d = max(degP, degQ). 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 ε-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 − εQ(x, y), ẏ = ∂H ∂x + εP (x, y). (1.1) An oval γ of the level curveH(x, y) = h which is a closed (but non-isolated) periodic trajectory for ε = 0, may generate a limit cycle for small nonzero values of ε only if the accumulated energy dissipation is zero in the first approximation, i.e., when