Properties on the Unit Circle of Polynomials with Unimodular Coefficients

Abstract

A concrete explicit construction of a unimodular polynomial with prescribed zeros on the unit circle is given. More precisely a polynomial P(z) = α₀ + α₁ z +…α _N z ^N is produced for which ∣α _i ∣ = 1 for all i = 0,1,…, N and for which P(α _j ) = 0 for a given set of α _j ,j= 1,2,…, n, ∣α _j ∣ = 1, and P(z) ≠ 0 elsewhere on ∣z∣ = 1. It is further shown how to extend this construction so as to maintain these properties and force the maximum of ∣P(z)∣ to occur at any given number β ≠ α _j ,j = 1,2,…, n and ∣β∣ = 1. The dependence of N on n is exponential, but there is reason to believe that this is actually necessary and not just a weakness of the method.

Also to appear in the Proceedings of the American Math Society.