On the Number of Solutions of Polynomial Congruences and Thue Equations

has only finitely many solutions in integers x and y. In the first part of this paper we shall establish upper bounds for the number of solutions of (1) in coprime integers x and y under the assumption that the discriminant D(F) of F is nonzero. For most integers h these bounds improve upon those obtained by Bombieri and Schmidt in [5]. In the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of Nagell [30], Ore [32], Sandor [33], and Huxley [19]. In fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is, in general, best possible. In the second part we shall address the problem of finding forms F for which (1) has many solutions for arbitrarily large integers h. Finally we shall obtain upper bounds for the number of solutions of certain Thue-Mahler and Ramanujan-Nagell equations by appealing to estimates of Evertse, Gyory, Stewart, and Tijdeman [17] for the number of solutions of S-unit equations.