Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

Abstract

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 } \{H(x,y)=\operatorname {const}\} over which the integral of a polynomial 1-form P ( x , y ) d x + Q ( x , y ) d y 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 n=\deg H and d = max ( deg ⁡ P , deg ⁡ Q ) 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 n and d d for the particular case of hyperelliptic polynomials H ( x , y ) = y 2 + U ( x ) H(x,y)=y^2+U(x) under the additional assumption that all critical values of U 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.