THE MAXIMAL VALUE OF POLYNOMIALS WITH RESTRICTED COEFFICIENTS Dubicks, Arturas; Jankauskas, Jonas;
Let be a fixed complex number. In this paper, we study the quantity , where is the set of all real polynomials of degree at most n-1 with coefficients in the interval [0, 1]. We first show how, in principle, for any given and , the quantity S(, n) can be calculated. Then we compute the limit for every of modulus 1. It is equal to 1/ if is not a root of unity. If , where and k [1, d-1] is an integer satisfying gcd(k, d) = 1, then the answer depends on the parity of d. More precisely, the limit is 1, 1/(d sin(/d)) and 1/(2d sin(/2d)) for d = 1, d even and d > 1 odd, respectively.
Newman polynomial;maximum of a polynomial;root of unity;Dirichlet's theorem;