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- W2998476032 abstract "Solving the equation $P_a(X):=X^{q+1}+X+a=0$ over 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,HELLESETH2008} and to construct error-correcting codes cite{Bracken2009}, 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}, 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 identify all the $GF{Q}$-zeros, even in the case $p=2$, it was solved only under the condition $gcd(n, k)=1$ cite{KM2019}. We discuss this equation without any restriction on $p$ and $gcd(n,k)$. New criteria for the number of the $GF{Q}$-zeros of $P_a(x)$ are proved. For the cases of one or two $GF{Q}$-zeros, we provide explicit expressions for these rational zeros in terms of $a$. For the case of $p^{gcd(n, k)}+1$ rational zeros, we provide a parametrization of such $a$'s and express the $p^{gcd(n, k)}+1$ rational zeros by using that parametrization." @default.
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- W2998476032 date "2019-12-29" @default.
- W2998476032 modified "2023-09-30" @default.
- W2998476032 title "Solving $X^{q+1}+X+a=0$ over Finite Fields" @default.
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