Perhaps you can assist me with some intuition. I'm looking for the number of zeros of

, where

and

are nonconstant polynomials with no common zeros, and

are nonnegative integers. As an example, the case

yields

, which has thirty zeros (the zeros of the polynomial in the numerator).

For some convenient notation, let

and

, where

is a complex number. I tried to decompose the rational function

in terms of its poles as follows:

Here, the

are complex numbers, and

is a polynomial of degree

. Specifically, if

and

(which is precisely when

does not vanish), we can write the terms in the right-hand side of equation

as a fraction with a common denominator, such that the polynomial

dominates the degree in its numerator. Hence,

has

zeros in this case.

There seem to be two more distinct cases, but I'm not sure how to prove what the number of zeros is in them. The answer in those (remaining) cases should (probably, based on my numerical experiments) be

if

and

if

and

Any ideas?

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