Abstract

A complex polynomial $H$ in two variables is called primitive of type $mathbb{C}^*$  if its generic fiber is biholomorphic to $mathbb{C}^*=mathbb{C} -{0}$ (for instance $H=x^2+y^2  mbox{ or }  H=x(xy-1) $).

Given an arbitrary primitive polynomial $H$ of type $mathbb{C}^*$ of degree $m+1$ and a polynomial 1-form $omega$ of degree  $n$ we provide an upper bound for the number of isolated zeros of the complete Abelian integral defined by $H$ and $omega$ depending only on  $m$ and $n$.

Work in progress.