The enumeration of simultaneous higher-order contacts between plane curves

TL;DR

This paper develops an intersection-theoretic formula using Semple bundle construction to count simultaneous higher-order contacts between generic plane curves and fixed curves, focusing on nonsingular points.

Contribution

It introduces a new formula for enumerating higher-order contacts between plane curves using advanced intersection theory techniques.

Findings

Derived an explicit formula for contact counts

Applicable to generic families of plane curves

Focuses on nonsingular intersection points

Abstract

Using the Semple bundle construction, we derive an intersection-theoretic formula for the number of simultaneous contacts of specified orders between members of a generic family of degree $d$ plane curves and finitely many fixed curves. The contacts counted by the formula occur at nonsingular points of both the members of the family and the fixed curves.