# On the intersection points of two plane algebraic curves

## TL;DR

This paper characterizes the set of intersection points of two plane algebraic curves of degrees m and n in complex plane, using conditions related to curves of specific degrees that contain all but one point of the set.

## Contribution

It provides necessary and sufficient conditions for a finite set of points to be the intersection of two algebraic curves, extending classical theorems with a new characterization.

## Key findings

- Conditions a) and b) are necessary and sufficient for the set to be intersection points.
- Conditions follow from Ceyley-Bacharach and Noether theorems.
- The characterization applies to complex plane algebraic curves.

## Abstract

We prove that a set $\mathcal X\subset \mathbb{C}^2,\ \#{\mathcal X}=mn,\ m\le n, $ is the set of intersection points of some two plane algebraic curves of degrees $m$ and $n,$ respectively, if and only if the following conditions are satisfied: a) Any curve of degree $m+n-3$ containing all but one point of $\mathcal X$, contains all of $\mathcal X,$ b) No curve of degree less than $m$ contains all of $\mathcal X.$ Let us mention that the conditions a) and b) in the "only if" direction of this result follow from the Ceyley-Bacharach and Noether theorems, respectively.