On the multiplicity of the hyperelliptic integrals

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 ${\mathbf{\text{C}}}^n$. 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 ${\mathbf{\text{C}}}^{n+1}$. We observe that $ordI(t)$ grows like an affine function with respect to $\mathrm{deg}\omega$.