On the equation x2l+1+x+a=0 over GF(2k)

In this paper, the polynomials Pa(x)=x2l+1+x+a with a GF(2k) are studied. Some new criteria for the number of zeros of Pa(x) in GF(2k) are proved. In particular, a criterion for Pa(x) to have exactly one zero in GF(2k) 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.