POLYNOMIALS WITH (0, -1) COEFFICIENTS AND A ROOT CLOSE TO A GIVEN POINT(1, )

For a fixed algebraic number alpha we discuss how closely a can be app roximated by a root of a (0, +1, -1) polynomial of given degree. We sh ow that the worst rate of approximation tends to occur for roots of un ity, particularly those of small degree. For roots of unity these boun ds depend on the order of vanishing, k, of the polynomial at alpha. In particular we obtain the following. Let B-N denote the set of roots o f all (0, +1, -1) polynomials of degree at most N and B-N(alpha, k) th e roots of those polynomials that have a root of order at most k at al pha. For a Pisot number alpha in (1,2] we show that min(beta)is an ele ment of B-N\{alpha} \alpha-beta\ = 1/alpha(N), and for a root of unity alpha that min(beta)is an element of B-N(alpha,k)\{alpha} \alpha-beta \ = 1/N(k+1[1/2 phi(d)]+1). We study in detail the case of alpha = 1, where, by far, the best approximations are real. We give fairly precis e bounds on the closest real root to 1. When k = 0 or 1 we can describ e the extremal polynomials explicitly.