Intersection Numbers and Split of Minor Roots

TL;DR

This paper investigates intersection numbers related to Jacobian pairs using roots analysis and explores root splits in a specific polynomial degree case, highlighting open problems in the field.

Contribution

It introduces new equations and inequalities for intersection numbers and analyzes root splits for degree (99, 66), proposing open questions.

Findings

Derived equations and inequalities for intersection numbers.

Analyzed root splits for degree (99, 66) case.

Identified an open problem in root splitting analysis.

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.