The P2 + P problem and conjectures of Pólya
Stephanie Edwards
Associate Professor
Department of Mathematics
Hope College
One of the problems stated in the Pólya and Szegö text from the early 1900's,
"Aufgaben und Lehrsätze aus der Analysis," is: If P is a real polynomial with only real zeros,
find the number of non-real zeros of P2 + 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 solve the P2 + P problem when P has only simple real zeros.
Further, we will show how the problem can be restated in terms of the number
of non-real zeros of the second derivative of a real entire function and discuss
the research and progress which has been made in the area of distribution of
zeros of real entire functions.