Complete hyperelliptic integrals of the first kind and their non-oscillation
| dc.creator | Gavrilov, Lubomir | |
| dc.creator | Iliev, Iliya D. | |
| dc.date | 2002-11-25 | |
| dc.date.accessioned | 2026-07-07T09:41:41Z | |
| dc.date.available | 2026-07-07T09:41:41Z | |
| dc.description | Let $P(x)$ be a real polynomial of degree $2g+1$, $H=y^2+P(x)$ and $δ(h)$ be an oval contained in the level set $\{H=h\}$. We study complete Abelian integrals of the form $$I(h)=\int_{δ(h)} \frac{(α_0+α_1 x+... + α_{g-1}x^{g-1})dx}{y}, h\in Σ,$$ where $α_i$ are real and $Σ\subset \R$ is a maximal open interval on which a continuous family of ovals $\{δ(h)\}$ 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 $δ(h)\subset\{H=h\}$, $h\inΣ$, such that the Abelian integral $I(h)$ can have at least $[\frac32g]-1$ zeros in $Σ$. Our main result is Theorem \ref{main} in which we show that when $g=2$, exceptional families of ovals $\{δ(h)\}$ exist, such that the corresponding vector space is still Chebyshev. | |
| dc.identifier | https://arxiv.org/abs/math/0211386 | |
| dc.identifier | http://arxiv.org/abs/math/0211386 | |
| dc.identifier | Trans. Amer. Math. Soc. vol. 356 (2004) 1185-1207 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161917 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 34C07; 3408; 70K05 | |
| dc.title | Complete hyperelliptic integrals of the first kind and their non-oscillation | |
| dc.type | text |