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 ) = a 0 + a 1 z + ⋯ a N z N P(z) = {a_0} + {a_1}z + \cdots {a_N}{z^N} is produced for which | a i | = 1 |{a_i}| = 1 for all i = 0 , 1 , … , N i = 0,1, \ldots ,N and for which P ( α j ) = 0 P({\alpha _j}) = 0 for a given set of α j , j = 1 , 2 , … , n , | α j | = 1 {\alpha _j},j = 1,2, \ldots ,n,|{\alpha _j}| = 1 , and P ( z ) ≠ 0 P(z) \ne 0 elsewhere on | z | = 1 |z| = 1 . It is further shown how to extend this construction so as to maintain these properties and force the maximum of | P ( z ) | |P(z)| to occur at any given number β ≠ α j , j = 1 , 2 , … , n \beta \ne {\alpha _j},j = 1,2, \ldots ,n and | β | = 1 |\beta | = 1 . The dependence of N N on n n is exponential, but there is rėason to believe that this is actually necessary and not just a weakness of the method.