Dmitry Novikov

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,

Redundant Picard–Fuchs System for Abelian Integrals

Abstract We derive an explicit system of Picard–Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants appears only in dimension approximately two times greater than the standard Picard–Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were able to majorize the number of all (real and complex) zeros. In the second part of the paper an equivariant formulation of the above problem is discussed and relationships between spread of critical values and non-homogeneity of uni- and bivariate complex polynomials are studied.

Meandering of trajectories of polynomial vector fields in the affine

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in R n and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated.

n

An elementary example shows that the number of zeroes of a component of a solution of a system of linear ordinary differential equations cannot be estimated through the norm of coefficients of the system. Bounds for oscillations. In [1] it was shown that a linear ordinary differential equation of order n, with real analytic coefficients bounded in a neighborhood of the interval [−1, 1], admits a uniform upper bound for the number of isolated zeros of a solution defined on this interval. The analyticity condition can be relaxed; only the boundedness of the coefficients matters. Probably, the simplest result in this spirit is the following theorem for the linear ordinary differential equation y(t) + a1(t)y(t) + · · ·+ an(t)y(t) = 0 (1) with continuous coefficients on [α, β] ⊂ R. Theorem 1 ([3, 4]). If the coefficients of the differential equation (1) are uniformly bounded by the constant C ≥ 1 (that is, max{|ai(t)| : i = 1, . . . , n} ≤ C), then a solution defined on [α, β] cannot have more than n−1 + n ln 2C|β−α| isolated zeros. An analog of this result for a system of ordinary differential equations, viewed as a vector field in space, would concern the number of isolated intersections between integral trajectories of the vector field and hyperplanes (or, more generally, hypersurfaces). For polynomial systems of degree d on R of the form ẋi = vi(t, x), i = 1, . . . , n, vi(t, x) = ∑ k+|α|≤d vikαt x, (2) and algebraic hypersurfaces given by {P = 0} where P = P (t, x) is a polynomial of degree d, the following theorem, proved in [3] (see also [2]), gives a bound for the number of isolated intersections in case the magnitude of the domain of the solution and the amplitude of the solution are controlled by the height of the polynomial system, that is, the number max{|vikα| : k + |α| ≤ d, i = 1, . . . , n}. Received by the editors July 31, 2000 and, in revised form, September 11, 2000. 1991 Mathematics Subject Classification. Primary 34C10, 34M10; Secondary 34C07.

-space

Let

Systems of linear ordinary differential equations with bounded coefficients may have very oscillating solutions

\Pi

On the finite cyclicity of open period annuli

be an open, relatively compact period annulus of real analytic vector field

The Pila–Wilkie theorem for subanalytic families: a complex analytic approach

X_0

Multiplicities of Noetherian Deformations

on an analytic surface. We prove that the maximal number of limit cycles which bifurcate from