Intersection Numbers and Split of Minor Roots

Abstract: There are two main parts in this manuscript. First, for a Jacobian pair $(f, g)$, with the concept of final major roots and final minor roots, we obtain equations and inequalities for intersection numbers ${\rm I}(f_\xi, g)$ and ${\rm I}(f_\xi, f_y)$ respectively, here, $\xi$ is a generic element of the base field and $f_\xi=f-\xi$ and discuss usage of them in some special cases. Second, we discuss all possibilities of the splits of principle minor roots for the case of degree (99, 66) with help of Abhyankar-Moh planar semigroup, find an unknown possible split and suggest case (99, 66) is open.