Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

Novikov, D.¹ ; Yakovenko, S.²

¹ Laboratoire de Topologie, Université de Bourgogne, Dijon, France
² Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, Israel

Electronic research announcements of the American Mathematical Society, Tome 05 (1999), pp. 55-65.

Résumé

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.

Détail

DOI : 10.1090/S1079-6762-99-00061-X

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