Karl Dilcher, Dalhousie University
"Zeros and irreducibility of gcd-polynomials"
The main object of this talk is a family of self-inversive polynomials of degree n whose j-th coefficient is gcd(n,j)k, for a fixed integer k\geq 1. I will show that these polynomials have all of their zeros on the unit circle, with uniform angular distribution. In the process I prove some new results on Jordan's totient function. I also show that these polynomials are irreducible, apart from an obvious linear factor, whenever nis a power of a prime, and conjecture that this holds for all n. Finally, I show how these methods can be used 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.)
MC 5417