On Two-Parametric Quartic Families of Diophantine Problems

Let n ≥ 3, v1(x), . . . , vn−1(x) ∈ Z[x] and u ∈ {−1, 1}, then x(x− v1(a)y) · · · (x− vn−1(a)y) + uy = ±1 is called a parametrized familiy of Thue equations, if a ∈ Z and the solutions x, y are searched in Z; cf. Thomas (1993). There are several results concerning parametrized families of cubic and quartic families of Thue equations, see Mignotte et al. (1996) and the references therein. Thomas (1993) proved that if 0 < b < c and a ≥ [2 × 10(b + 2c)] 4.85 c−b , then the only solutions of the equation x(x− ay)(x− ay) + uy = ±1 u = ±1 are ±{(1, 0), (0, u), (au, u), (au, u)}. As far as we know this is the only case, when a two-parametric familiy of Thue equations is completely solved. In this paper we consider diophantine problems which are related to a two-parametric quartic polynomial. Let 1 < a < b, a, b ∈ Z and Pa,b(x) = x(x− 1)(x− a)(x− b)− 1. This polynomial is irreducible (see Halter-Koch et al. (1998)). Denoting by α one of the zeros of Pa,b(x) the number field K = Q(α) is quartic. Let O = Z[α] and UO be the group of units of O. In the first part, we study the algebraic properties of K. We prove