# Some Results on Linearized Trinomials that Split Completely

## TL;DR

This paper investigates the properties of linearized polynomials over finite fields, focusing on calculating their rank and nullity, and characterizes certain trinomials that split completely for specific field sizes.

## Contribution

It provides a new method to determine the nullity of linearized polynomials using associated matrices and characterizes splitting trinomials for certain parameters.

## Key findings

- Nullity of linearized polynomials can be computed via associated matrices.
- Characterization of splitting trinomials for field sizes up to d^2 - d + 1.
- Connections established between polynomial properties and finite geometry.

## Abstract

Linearized polynomials over finite fields have been much studied over the last several decades. Recently there has been a renewed interest in linearized polynomials because of new connections to coding theory and finite geometry. We consider the problem of calculating the rank or nullity of a linearized polynomial $L(x)=\sum_{i=0}^{d}a_i x^{q^i}$ (where $a_i\in \mathbb{F}_{q^n}$) from the coefficients $a_i$. The rank and nullity of $L(x)$ are the rank and nullity of the associated $\mathbb{F}_q$-linear map $\mathbb{F}_{q^n} \longrightarrow \mathbb{F}_{q^n}$. McGuire and Sheekey defined a $d\times d$ matrix $A_L$ with the property that $$\mbox{nullity} (L)=\mbox{nullity} (A_L -I).$$ We present some consequences of this result for some trinomials that split completely, i.e., trinomials $L(x)=x^{q^d}-bx^q-ax$ that have nullity $d$. We give a full characterization of these trinomials for $n\le d^2-d+1$.