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Number of zeros of a certain type of rational function
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Posted 2017-08-17, 08:31 AM
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?
"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram

Last edited by Chruser; 2017-10-07 at 07:30 AM.
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