Lacunary polynomials, multiple blocking sets and Baer subplanes

New lower bounds are given for the size of a point set in a Desarguesian pr ojective plane over a finite field that contains at least a prescribed numb er s of points on every line. These bounds are best possible when q is squa re and s is small compared with q. In this case the smallest set is shown t o be the union of disjoint Baer subplanes. The results are based on new res ults on the structure of certain lacunary polynomials, which can be regarde d as a generalization of Redei's results in the case when the derivative of the polynomial vanishes.