Discriminant, vertex and general type of a family of polynomials

TL;DR

This paper develops a formal framework for understanding the limits of dual plane curves using Katz's method, providing formulas for vertices of plane curves in a family and showing their dependence on initial terms and general type.

Contribution

It introduces a formal expression for limits of dual plane curves and relates vertices to the initial terms and the general type of the family.

Findings

Vertices depend only on the first τ terms of the family

Provides a formula for vertices of plane curves in a family

Uses Katz's method to analyze limits of dual curves

Abstract

In this paper we give a formal expression to the limits of dual plane curves by using the method introduced by Katz. As an application, we give a formula to compute the vertices of $C(0)$ a plane curve in $C(t)$ a one-parameter family of plane curves and we show that those vertices depend only on the first $\tau$ terms of that family, where $\tau$ is the general type of $C(t)$.