# Counting the number of roots of a polynomial in each quadrant of the complex plane

I'm looking to answer the question:

Given a polynomial of a single variable $$x$$: $$\sum_n a_n x^n = 0$$, how many roots are there in each quadrant of the complex plane, counting positive/negative real/imaginary lines seperately?

I don't want to solve this numerically if I don't have to. This seems to me like it would be an ideal application of some kind of abstract algebra, except I don't know any? Can someone point me to a field which studies this problem - if it exists? How hard is this problem to solve?

On Wikipedia I've found Real Root Isolation, Descarte's Rule of Signs, and Sturm's Theorem. The one problem I have with Sturm's theorem is that I would like to find simple criteria, that can be worked out on paper, but this would seem to require a computer. Is there anything else that would help me solve this problem?

One can also think about dividing the real parameter space $$(a_0, ... ,a_n)$$ into regions with a certain number of real roots by exploiting the fact that the number of real roots changes when two complex roots come together (forming a multiple root). The value of the discriminant of a univariate polynomial encodes some of this information, since it is zero exactly when a univariate polynomial has a multiple root. When the degree is four or higher, the discriminant is positive if and only if the number of non-real roots is a multiple of four.
As for classifying roots in other quadrants of the complex plane, I'm not entirely sure, but it's a very interesting one. One might be able to take advantage of the work on counting real solutions. For example, if one appends the equation $$x^2-y=0$$, one can then ask how many real solutions in $$y$$ are there, giving the count of purely imaginary plus real solutions.