On polynomial congruences

On the one hand, we extend an estimate of Loxton and Smith on the number of solutions to a polynomial congruence in one unknown. More precisely, our work extends a result of Chalk who gave a precise description of the solutions to polynomial congruences f = 0 mod p^s for large s > C(f). On the other hand, we extend work of Stewart who gave a precise description of the above congruences for all s \ge 1 in the case where the discriminant of f does not vanish. We treat the general case where the discriminant may or may not vanish.