Rational Points on Certain Hyperelliptic Curves over Finite Fields

Abstract

Abstract. Let K be a field, a, b ∈ K and ab ̸ = 0. Let us consider the polynomials g1(x) = x n + ax + b, g2(x) = x n + ax 2 + bx, where n is a fixed positive integer. In this paper we show that for each k ≥ 2 the hypersurface given by the equation S i k: u 2 kY = gi(xj), i = 1, 2. j=1 contains a rational curve. Using the above and Woestijne’s recent results [8] we show how one can construct a rational point different from the point at infinity on the curves Ci: y 2 = gi(x), (i = 1, 2) defined over a finite field, in polynomial time. Dedicated to the memory of Andrzej M¸akowski 1.