Counting higher order tangencies for plane curves

TL;DR

This paper generalizes a previous result by establishing an upper bound on the number of points where $k$-th order tangencies occur among $n$ plane algebraic curves, extending understanding of their intersection properties.

Contribution

It introduces a bound on $k$-th order tangencies among plane algebraic curves, extending earlier work on first order tangencies.

Findings

Bound of $O(n^{(k+2)/(k+1)})$ points for $k$-th order tangencies

Generalization of previous first order tangency results

Provides a framework for analyzing higher order curve interactions

Abstract

We prove that $n$ plane algebraic curves determine $O(n^{(k+2)/(k+1)})$ points of $k$-th order tangency. This generalizes an earlier result of Ellenberg, Solymosi, and Zahl on the number of (first order) tangencies determined by $n$ plane algebraic curves.