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 Abstract: 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). [Permanent link]