On the curve Y n = X m + X over finite fields

We show that a maximal curve over Fq2 defined by the affine equation y n = f (x), where f (x) ∈ Fq2 [x] has degree coprime to n, is such that n is a divisor of q + 1 if and only if f (x) has a root in Fq2 . In this case, all the roots of f (x) belong to Fq2 ;cf. Thm. 1.2, Thm. 4.3 in [J. Pure Appl. Algebra 212 (2008), 2513–2521]. In particular, we characterize certain maximal curves defined by equations of type y n = xm + x over finite fields.