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

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 $$I(h)={\int }_{\delta (h)}\frac{({\alpha }_0+{\alpha }_{1}x+...+{\alpha }_{g-1}{x}^{g-1})dx}{y},h\in \Sigma ,$$ 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 $[\frac{3}{2}g]-1$ zeros in $\Sigma$. Our main result is Theorem \ref{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.