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 $\{H(x,y)=\operatorname{const}\}$ 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 $H(x,y)=y^2+U(x)$ 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.