Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

Novikov, D.; Yakovenko, S.

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

Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian systems

Authors: D. Novikov and S. Yakovenko
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 55-65
MSC (1991): Primary 14K20, 34C05, 58F21; Secondary 34A20, 30C15
DOI: https://doi.org/10.1090/S1079-6762-99-00061-X
Published electronically: April 30, 1999
MathSciNet review: 1679454
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Abstract: The tangential Hilbert 16th problem is to place an upper bound for the number of isolated ovals of algebraic level curves $\{H(x,y)=\operatorname {const}\}$ 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.

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