“Where have all the zeros gone?” By Stephanie Edwards, Mathematics and Statistics Department, Hope College

Many open problems in entire function theory, specifically, the distribution of zeros of real entire functions, can be traced back to work by George Pólya and Carl Gauss. One of the problems stated in a Pólya and Szegö text from the early 1900’s is: If P is a real polynomial with only real zeros, find the number of non-real zeros of P 2 + P′. If one removes the hypothesis that P has only real zeros, the problem becomes quite hard and was not solved until the 1980’s. We will discuss a simple solution to the P 2 + P′ problem, look at natural questions that arise from the problem, and discuss some open questions that have their roots in Pólya and Gauss.