Elementary Symmetric Polynomials in Numbers of Modulus 1

Abstract

AbstractWe describe the set of numbers σk(z1,…,zn+1), where z1,…,zn+1 are complex numbers of modulus 1 for which z1z2 … zn+1 = 1, and σk denotes the k-th elementary symmetric polynomial. Consequently, we give sharp constraints on the coefficients of a complex polynomial all of whose roots are of the same modulus. Another application is the calculation of the spectrum of certain adjacency operators arising naturally on a building of type Ãn.