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Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian
systems
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## Tangential Hilbert problem for perturbations of hyperelliptic Hamiltonian
systems

### D. Novikov, S. Yakovenko

**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.

*Copyright 1999 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**05** (1999), pp. 55-65
- Publisher Identifier: S 1079-6762(99)00061-X
- 1991
*Mathematics Subject Classification*. Primary 14K20, 34C05, 58F21
*Key words and phrases*.
- Received by the editors October 23, 1998
- Posted on April 30, 1999
- Communicated by Jeff Xia
- Comments (When Available)

**D. Novikov**

Laboratoire de
Topologie, Universit\'e de Bourgogne, Dijon, France

*E-mail address:* `novikov@topolog.u-bourgogne.fr`

**S. Yakovenko**

Department of Theoretical Mathematics,
The Weizmann Institute of Science, Rehovot, Israel

*E-mail address:* `yakov@wisdom.weizmann.ac.il`

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