On additive polynomials and certain maximal curves (Q942206)

It is an important problem to study maximal curves over finite fields. In this interesting paper the authors study a class of maximal curves with a specified affine equation. Let \(k\) be a field of positive characteristic \(p\) and \(A(x)\) an additive and separable polynomial in \(k[X]\). The authors show that all roots of \(A(x)\) belong to \(\mathbb{F}_{q^2}\) if the curve \(\mathcal{C}\) over \(\mathbb{F}_{q^2}\) given by \(A(X)=F(Y)\) is maximal where \(F(Y)\in \mathbb{F}_{q^2}[Y]\) is a polynomial of degree \(m\) prime to \(p\). In addition, they prove that \(m\) divides \(q+1\) in the special case \(F(Y)=Y^m\). In the paper they use various methods including \(p\)-adic Newton polygons of some curves.