An explicit construction for $n$-contact curves to a smooth cubic via divisions of polynomials and Zariski tuples

Abstract

Let $E$ be a smooth cubic. A plane curve $D$ is said to be an $n$-contact curve to $E$ if the intersection multiplicities at each intersection point between $E$ and $D$ is $n$. In this note, we give an algorithm to produce possible candidates for $n$-contact curves to $E$ and consider its application.