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 \[ I(h)=\int _{\delta (h)} \frac {(\alpha _0+\alpha _1 x+\ldots + \alpha _{g-1}x^{g-1})dx}{y}, \;\;h\in \Sigma , \] 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 $[\frac 32g]-1$ 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.