Maciej Ulas
(2007)
Bulletin of the Polish Academy of Sciences. Mathematics
Let K be a field, a,b ∈ K and ab ≠ 0. Consider the polynomials g₁(x) = xⁿ+ax+b, g₂(x) = xⁿ+ax²+bx, where n is a fixed positive integer. We show that for each k≥ 2 the hypersurface given by the equation
, i=1,2,
contains a rational curve. Using the above and van de Woestijne’s recent results we show how to construct a rational point different from the point at infinity on the curves , (i=1,2) defined over a finite field, in polynomial time.