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

Author(s)

Borges, Herivelto ^[1] ; Coutinho, Mariana ^[1]

Affiliation

^[1] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Ave Trabalhador Sao Carlense 400, BR-13566590 Sao Carlos, SP - Brazil

• Document type: Journal article
• Source: Journal of Pure and Applied Algebra; v. 224, n. 1, p. 239-249, JAN 2020.
• Web of Science Citations: 0

Abstract

Let g be the projective plane curve defined over F-q given by aX(n)Y(n) - X(n)Z(n) - Y(n)Z(n) + bZ(2n) = 0, where ab is not an element of [0, 1], and for each s is an element of [2, ,n - 1], let D-s(P1,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 P-1 = (1 : 0 : 0) and P-2 = (0 : 1 : 0) of g. The present work determines an upper bound for the number N-q (g) of F-q-rational points on the nonsingular model of g in cases where D-s(P1,P2) is F-q-Frobenius classical. As a consequence, when F-q is a prime field, the bound obtained for N-q (g) improves in several cases the known bounds for the number n(P) of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point P distinct from its vertices. (C) 2019 Elsevier B.V. All rights reserved. (AU)

• FAPESP's process: 17/04681-3 - Algebraic curves over finite fields
• Grantee: Herivelto Martins Borges Filho
• Support Opportunities: Regular Research Grants