On some generalized Fermat curves and chords of an affinely regular polygon inscribed in a hyperbola

Abstract

Let G be the projective plane curve defined over Fq given byaXnYn−XnZn−YnZn+bZ2n=0, where ab∉{0,1}, and for each s∈{2,…,n−1}, let DsP1,P2 be the base-point-free linear series cut out on G by the linear system of all curves of degree s passing through the singular points P1=(1:0:0) and P2=(0:1:0) of G. The present work determines an upper bound for the number Nq(G) of Fq-rational points on the nonsingular model of G in cases where DsP1,P2 is Fq-Frobenius classical. As a consequence, when Fq is a prime field, the bound obtained for Nq(G) improves in several cases the known bounds for the number nP of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point P distinct from its vertices.