Lacunary Polynomials, Multiple Blocking Sets and Baer Subplanes

New lower bounds are given for the size of a point set in aDesarguesian projective plane over a finite field that containsat least a prescribed number s of points on every line. Thesebounds are best possible when q is square and s is small comparedwith q. In this case the smallest set is shown to be the unionof disjoint Baer subplanes. The results are based on new resultson the structure of certain lacunary polynomials, which canbe regarded as a generalization of Rédei's results inthe case when the derivative of the polynomial vanishes.