# Planar Polynomials arising from Linearized polynomials

## TL;DR

This paper constructs and classifies a family of planar polynomials over finite fields, revealing their structure through algebraic curves and providing a comprehensive understanding of their properties.

## Contribution

It introduces a new class of planar polynomials derived from linearized polynomials and fully classifies the parameter pairs that produce planar functions.

## Key findings

- Complete classification of pairs (A,B) for planarity
- Connection between planar polynomials and algebraic curves
- New family of planar polynomials over finite fields

## Abstract

In this paper we construct planar polynomials of the type $f_{A,B}(x)=x(x^{q^2}+Ax^{q}+Bx)\in \mathbb{F}_{q^3}[x]$, with $A,B \in \mathbb{F}_{q}$. In particular we completely classify the pairs $(A,B)\in \mathbb{F}_{q}^2$ such that $f_{A,B}(x)$ is planar using connections with algebraic curves over finite fields.