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29 March 2024
 
  » arxiv » math.DS/0211386

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Complete hyperelliptic integrals of the first kind and their non-oscillation
Lubomir Gavrilov ; Iliya D. Iliev ;
Date 25 Nov 2002
Subject Dynamical Systems MSC-class: 34C07; 3408; 70K05 | math.DS
AbstractLet $P(x)$ 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 $$I(h)=int_{delta(h)} frac{(alpha_0+alpha_1 x+... + alpha_{g-1}x^{g-1})dx}{y}, hin Sigma,$$ where $alpha_i$ are real and $Sigmasubset R$ 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 $delta(h)subset{H=h}$, $hinSigma$, such that the Abelian integral $I(h)$ can have at least $[frac32g]-1$ zeros in $Sigma$. Our main result is Theorem ef{main} in which we show that when $g=2$, exceptional families of ovals ${delta(h)}$ exist, such that the corresponding vector space is still Chebyshev.
Source arXiv, math.DS/0211386
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