On the multiplicity of the hyperelliptic integrals

Abstract

Let $I(t)= \oint_{\delta(t)} \omega$ be an Abelian integral, where $H=y^2-x^{n+1}+P(x)$ is a hyperelliptic polynomial of Morse type, $\delta(t)$ a horizontal family of cycles in the curves $\{H=t\}$, and $\omega$ a polynomial 1-form in the variables $x$ and $y$. We provide an upper bound on the multiplicity of $I(t)$, away from the critical values of $H$. Namely: $ord\ I(t) \leq n-1+\frac{n(n-1)}{2}$ if $\deg \omega <\deg H=n+1$. The reasoning goes as follows: we consider the analytic curve parameterized by the integrals along $\delta(t)$ of the $n$ ``Petrov'' forms of $H$ (polynomial 1-forms that freely generate the module of relative cohomology of $H$), and interpret the multiplicity of $I(t)$ as the order of contact of $\gamma(t)$ and a linear hyperplane of $\textbf C^ n$. Using the Picard-Fuchs system satisfied by $\gamma(t)$, we establish an algebraic identity involving the wronskian determinant of the integrals of the original form $\omega$ along a basis of the homology of the generic fiber of $H$. The latter wronskian is analyzed through this identity, which yields the estimate on the multiplicity of $I(t)$. Still, in some cases, related to the geometry at infinity of the curves $\{H=t\} \subseteq \textbf C^2$, the wronskian occurs to be zero identically. In this alternative we show how to adapt the argument to a system of smaller rank, and get a nontrivial wronskian. For a form $\omega$ of arbitrary degree, we are led to estimating the order of contact between $\gamma(t)$ and a suitable algebraic hypersurface in $\textbf C^{n+1}$. We observe that $ord I(t)$ grows like an affine function with respect to $\deg \omega$.