Albion College

Mathematics and Computer Science

Mathematics and Computer Science

COLLOQUIUM

The *P*^{2} + *P* problem and conjectures of Pólya

Stephanie Edwards

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 *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 solve the *P*^{2} + *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.