On the number of rational points on Artin-Schreier hypersurfaces

Author(s)

Oliveira, Jose Alves ; Borges, Herivelto ; Martinez, F. E. Brochero

• Document type: Journal article
• Source: FINITE FIELDS AND THEIR APPLICATIONS; v. 90, p. 25-pg., 2023-05-15.

Abstract

Let Fqk denote the finite field with qk elements. In this work, we study the number of Fqk-rational points of the affine hypersurfaces given by yq - y = a1xd11 + center dot center dot center dot + asxds s + b. We prove that if b is an element of Fqk has a nonzero trace over Fq, then Weil's bound cannot be achieved and that a sharp bound can be explicitly provided. In the case Trqk/q(b) = 0, we give necessary and sufficient conditions for Weil's bound to be attained. Furthermore, we present several new cases in which formulas for the number of Fqk-rational points can be obtained.(c) 2023 Elsevier Inc. All rights reserved. (AU)

• FAPESP's process: 22/15301-5 - Algebraic curves over finite fields and applications
• Grantee: Herivelto Martins Borges Filho
• Support Opportunities: Regular Research Grants
• FAPESP's process: 21/13712-5 - Special Elements over finite fields
• Grantee: José Alves Oliveira
• Support Opportunities: Scholarships in Brazil - Post-Doctoral