2019/12/29 by Kim, Kwang Ho, Choe, Junyop, Mesnager, Sihem
#FOS: Computer and information sciences #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.1912.12648
Solving the equation Pa(X):=Xq+1+X+a=0 over finite field \GFQ, where Q=pn, q=pk and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem \citeACZ2000, the construction of difference sets with Singer parameters \citeDD2004, determining cross-correlation between m-sequences \citeDOBBERTIN2006,HELLESETH2008 and to construct error-correcting codes \citeBracken2009, as well as to speed up the index calculus method for computing discrete logarithms on finite fields \citeGGGZ2013,GGGZ2013+ and on algebraic curves \citeM2014. Subsequently, in \citeBluher2004,HK2008,HK2010,BTT2014,Bluher2016,KM2019,CMPZ2019,MS2019, the \GFQ-zeros of Pa(X) have been studied: in \citeBluher2004 it was shown that the possible values of the number of the zeros that Pa(X) has in \GFQ is 0, 1, 2 or pgcd(n, k)+1. Some criteria for the number of the \GFQ-zeros of Pa(x) were found in \citeHK2008,HK2010,BTT2014,KM2019,MS2019. However, while the ultimate goal is to identify all the \GFQ-zeros, even in the case p=2, it was solved only under the condition gcd(n, k)=1 \citeKM2019. We discuss this equation without any restriction on p and gcd(n,k). New criteria for the number of the \GFQ-zeros of Pa(x) are proved. For the cases of one or two \GFQ-zeros, we provide explicit expressions for these rational zeros in terms of a. For the case of pgcd(n, k)+1 rational zeros, we provide a parametrization of such a's and express the pgcd(n, k)+1 rational zeros by using that parametrization.