English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The problem of solving explicitly the equation $P_a(X):=X^{q+1}+X+a=0$ overthe finite field $\GF{Q}$, where $Q=p^n$, $q=p^k$ and $p$ is a prime, arises inmany 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 tospeed up the index calculus method for computing discrete logarithms on finitefields \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 wasshown 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 theultimate 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 toexplicitly compute the zeros. This paper solves the remained problem, thus nowthe equation $X^{p^k+1}+X+a=0$ over $\GF{p^n}$ is completely solved for anyprime $p$, any integers $n$ and $k$.

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