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 | References | Similar Articles | Additional Information

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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Additional Information

**D. Novikov**

Affiliation:
Laboratoire de Topologie, Université de Bourgogne, Dijon, France

Email:
novikov@topolog.u-bourgogne.fr

**S. Yakovenko**

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

Received by editor(s):
October 23, 1998

Published electronically:
April 30, 1999

Communicated by:
Jeff Xia

Article copyright:
© Copyright 1999
American Mathematical Society