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Complete hyperelliptic integrals of the first kind and their non-oscillation

Authors: Lubomir Gavrilov and Iliya D. Iliev
Journal: Trans. Amer. Math. Soc. 356 (2004), 1185-1207
MSC (2000): Primary 34C08; Secondary 14D05, 14K20, 34C07
Published electronically: September 22, 2003
MathSciNet review: 2021617
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Abstract: Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $\delta(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form

\begin{displaymath}I(h)=\int_{\delta(h)} \frac{(\alpha_0+\alpha_1 x+\ldots + \alpha_{g-1}x^{g-1})dx}{y}, \;\;h\in \Sigma, \end{displaymath}

where $\alpha_i$ are real and $\Sigma\subset \mathbb{R}$ is a maximal open interval on which a continuous family of ovals $\{\delta(h)\}$ exists. We show that the $g$-dimensional real vector space of these integrals is not Chebyshev in general: for any $g>1$, there are hyperelliptic Hamiltonians $H$ and continuous families of ovals $\delta(h)\subset\{H=h\}$, $h\in\Sigma$, such that the Abelian integral $I(h)$ can have at least $[\frac32g]-1$ zeros in $\Sigma$. Our main result is Theorem 1 in which we show that when $g=2$, exceptional families of ovals $\{\delta(h)\}$ exist, such that the corresponding vector space is still Chebyshev.

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Additional Information

Lubomir Gavrilov
Affiliation: Laboratoire Emile Picard, CNRS UMR 5580, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France

Iliya D. Iliev
Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria

Received by editor(s): December 18, 2002
Published electronically: September 22, 2003
Article copyright: © Copyright 2003 American Mathematical Society

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