- CARMA COLLOQUIUM
- Speaker: Karl Dilcher, Mathematics and Statistics, Dalhousie University
- Title: Zeros and irreducibility of gcd-polynomials
- Location: Room V205, Mathematics Building (Callaghan Campus) The University of Newcastle
- Time and Date: 4:00 pm, Thu, 15th Oct 2015
We study the family of self-inversive polynomials of degree $n$ whose $j$th
coefficient is $\gcd(n,j)^k$, for a fixed integer $k \geq 1$. We prove
that these polynomials have all of their roots on the unit circle, with
uniform angular distribution. In the process we prove some new results
on Jordan's totient function. We also show that these polynomials
are irreducible, apart from an obvious linear factor, whenever $n$ is a
power of a prime, and conjecture that this holds for all $n$.
Finally we use some of these methods to obtain general results on the zero
distribution of self-inversive polynomials and of their "duals" obtained
from the discrete Fourier transforms of the coefficients sequence.
(Joint work with Sinai Robins).
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