Preimages of $p-$Linearized Polynomials over $\GF{p}$

TL;DR

This paper investigates the explicit computation of preimages of certain $p$-linearized polynomials over finite fields, extending previous results to polynomials with coefficients in $ ext{GF}(p)$ and providing formulas for their preimages.

Contribution

It generalizes the explicit preimage computation of $p$-linearized polynomials over $ ext{GF}(p^n)$, focusing on those dividing $X - X^{p^k}$, with new formulas for these cases.

Findings

Explicit preimages over $ ext{GF}(p^n)$ for polynomials dividing $X - X^{p^k}$

Extension of previous results from $ ext{GF}(p^n)$ to polynomials over $ ext{GF}(p)$

New formulas for preimages of $p$-linearized polynomials with coefficients in $ ext{GF}(p)$

Abstract

Linearized polynomials over finite fields have been intensively studied over the last several decades. Interesting new applications of linearized polynomials to coding theory and finite geometry have been also highlighted in recent years. Let $p$ be any prime. Recently, preimages of the $p-$linearized polynomials $\sum_{i=0}^{\frac kl-1} X^{p^{li}}$ and $\sum_{i=0}^{\frac kl-1} (-1)^i X^{p^{li}}$ were explicitly computed over $\GF{p^n}$ for any $n$. This paper extends that study to $p-$linearized polynomials over $\GF{p}$, i.e., polynomials of the shape $$L(X)=\sum_{i=0}^t \alpha_i X^{p^i}, \alpha_i\in\GF{p}.$$ Given a $k$ such that $L(X)$ divides $X-X^{p^k}$, the preimages of $L(X)$ can be explicitly computed over $\GF{p^n}$ for any $n$.