TITLE: When does a polynomial have all pure imaginary roots?
QUESTION [6 upvotes]: Let $P(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}$, where $a_1, a_2, \dots, a_n$ are intergers.
Question 1. When does the polynomial $P(x)$ have its zeros all being pure imaginary or zero(here 0 is a root of the given polynomial)?
Question 2. Does there exist a characterization that $P(x)$ have its zeros all being pure imaginary or zero?
Please point me to some references if this has already been studied. Thanks for your time!

REPLY [3 votes]: There is a method due to Sturm which allows the determination of the number of roots of a polynomial in any real interval, in particular you may apply this to find the number of negative roots. The following Wikipedia article may serve as an initial reference:
http://en.wikipedia.org/wiki/Sturm%27s_theorem