On the curve yn=xm+xyn=xm+x over finite fields

We characterize certain maximal curves over finite fields defined by equations of type yn=xm+xyn=xm+x. Moreover, we show that a maximal curve over Fq2Fq2 defined by the affine equation yn=f(x)yn=f(x), where f(x)∈Fq2[x]f(x)∈Fq2[x] is separable of degree coprime to n, is such that n   is a divisor of q+1q+1 if and only if f(x)f(x) has a root in Fq2Fq2. In this case, all the roots of f(x)f(x) belong to Fq2Fq2; cf. Theorems 1.2 and 4.3 in Garcia and Tafazolian (2008) [9].