We describe the set of numbers sigma_k(z₁, ...,z_n+1), where z₁, ..., z_n+1 are complex numbers of modulus 1 for which z₁z₂ cdots z_n+1=1, and sigma_k 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 {\tilde A}_n.

MSC Classifications:

05E05, 33C45, 30C15, 51E24 show english descriptions Symmetric functions and generalizations
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
Buildings and the geometry of diagrams 05E05 - Symmetric functions and generalizations
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]
30C15 - Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
51E24 - Buildings and the geometry of diagrams

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