# Quadratic Maps in Two Variables on Arbitrary Fields

## TL;DR

This paper classifies homogeneous quadratic maps in two variables over arbitrary fields with characteristic not 2 or 3, providing a comprehensive framework and computational tools for recognizing equivalence under linear transformations.

## Contribution

It offers a generic classification of quadratic maps in two variables over arbitrary fields and introduces efficient criteria for recognizing their equivalence.

## Key findings

- Complete classification of quadratic maps in two variables
- Efficient computational criteria for equivalence recognition
- Applicable over fields with characteristic not 2 or 3

## Abstract

Let $\mathbb{F}$ be a field of characteristic different from $2$ and $3$, and let $V$ be a vector space of dimension $2$ over $\mathbb{F}$. The generic classification of homogeneous quadratic maps $f\colon V\to V$ under the action of the linear group of $V$, is given and efficient computational criteria to recognize equivalence are provided.