Complete hyperelliptic integrals of the first kind and their non-oscillation期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Lubomir Gavrilov Iliya D Iliev

Institution:	Laboratoire Emile Picard, CNRS UMR 5580, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France ; Institute of Mathematics, Bulgarian Academy of Sciences, P.O. Box 373, 1090 Sofia, Bulgaria

Abstract:	Let be a real polynomial of degree , 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 -dimensional real vector space of these integrals is not Chebyshev in general: for any , there are hyperelliptic Hamiltonians and continuous families of ovals $\delta(h)\subset\{H=h\}$ , $h\in\Sigma$ , such that the Abelian integral can have at least $\frac32g]-1$ zeros in $\Sigma$ . Our main result is Theorem 1 in which we show that when , exceptional families of ovals $\{\delta(h)\}$ exist, such that the corresponding vector space is still Chebyshev.

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