On the equation xl2+1+x+a=0 over GF(k2)

In this paper, the polynomials Pa(x)=xl2+1+x+a with a∈GF(k2) are studied. Some new criteria for the number of zeros of Pa(x) in GF(k2) are proved. In particular, a criterion for Pa(x) to have exactly one zero in GF(k2) when gcd(l,k)=1 is formulated in terms of the values of polynomials introduced by Dobbertin. In the case when there is a unique zero, this root is calculated explicitly.