Complete solution over $\GF{p^n}$ of the equation $X^{p^k+1}+X+a=0$

TL;DR

This paper provides a complete explicit solution for the equation $X^{p^k+1}+X+a=0$ over finite fields $ ext{GF}(p^n)$, regardless of prime $p$ or the relationship between $n$ and $k$, resolving a longstanding open problem.

Contribution

It offers the first explicit solutions for all cases of the equation over finite fields, removing previous restrictions on $p$ and $ ext{gcd}(n,k)$.

Findings

Explicit solutions for all zero counts of the equation.

Complete characterization of solutions over $ ext{GF}(p^n)$.

Resolution of the open problem for the case with $p^{ ext{gcd}(n,k)}+1$ zeros.

Abstract

The problem of solving explicitly the equation $P_a(X):=X^{q+1}+X+a=0$ over the finite field $\GF{Q}$, where $Q=p^n$, $q=p^k$ and $p$ is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \cite{ACZ2000}, the construction of difference sets with Singer parameters \cite{DD2004}, determining cross-correlation between $m$-sequences \cite{DOBBERTIN2006} and to construct error correcting codes \cite{Bracken2009}, cryptographic APN functions \cite{BTT2014,Budaghyan-Carlet_2006}, designs \cite{Tang_2019}, as well as to speed up the index calculus method for computing discrete logarithms on finite fields \cite{GGGZ2013,GGGZ2013+} and on algebraic curves \cite{M2014}. Subsequently, in \cite{Bluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019,KCM19}, the $\GF{Q}$-zeros of $P_a(X)$ have been studied. In \cite{Bluher2004}, it was shown that the possible values of the number of the zeros that $P_a(X)$ has in $\GF{Q}$ is $0$, $1$, $2$ or $p^{\gcd(n, k)}+1$. Some criteria for the number of the $\GF{Q}$-zeros of $P_a(x)$ were found in \cite{HK2008,HK2010,BTT2014,KM2019,MS2019}. However, while the ultimate goal is to explicit all the $\GF{Q}$-zeros, even in the case $p=2$, it was solved only under the condition $\gcd(n, k)=1$ \cite{KM2019}. In this article, we discuss this equation without any restriction on $p$ and $\gcd(n,k)$. In \cite{KCM19}, for the cases of one or two $\GF{Q}$-zeros, explicit expressions for these rational zeros in terms of $a$ were provided, but for the case of $p^{\gcd(n, k)}+1$ $\GF{Q}-$ zeros it was remained open to explicitly compute the zeros. This paper solves the remained problem, thus now the equation $X^{p^k+1}+X+a=0$ over $\GF{p^n}$ is completely solved for any prime $p$, any integers $n$ and $k$.