Paper 2020/160

Solving Some Affine Equations over Finite Fields

Sihem Mesnager, Kwang Ho Kim, Jong Hyok Choe, and Dok Nam Lee

Abstract

Let $l$ and $k$ be two integers such that $l|k$. Define $T_l^k(X):=X+X^{p^l}+\cdots +X^{p^{l(k/l-2)}}+X^{p^{l(k/l-1)}}$ and $S_l^k(X):=X-X^{p^l}+\cdots +(-1{)}^{(k/l-1)}{X}^{p^{l(k/l-1)}}$, where $p$ is any prime. This paper gives explicit representations of all solutions in $$ to the affine equations $$ and $$, $$. For the case $$ that was solved very recently in \cite{MKCL2019}, the result of this paper reveals another solution.