Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

The tangential Hilbert 16th problem is to place an upper bound for the numb er 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 i ntegral) 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 primit ive recursive (in fact, elementary) function of n and d for the particular case of hyperelliptic polynomials H(x; y) = y(2) + U(x) under the additiona l assumption that all critical values of U are real. This is the first gene ral 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 exp osition. The main ingredients of the proof are explained and the differenti al algebraic generalization (that is the core result) is given.