POLYNOMIALS WITH HEIGHT 1 AND PRESCRIBED VANISHING AT 1

PETER BORWEIN AND MICHAEL J. MOSSINGHOFF

Abstract. We study the minimal degree d(m) of a polynomial with all coefficients in {−1, 0, 1} and a zero of order m at 1. We determine d(m) for m ≤ 10 and compute all the extremal polynomials. We also determine the minimal degree for m = 11 and m = 12 among certain symmetric polynomials, and we find explicit examples with small degree for m ≤ 21. Each of the extremal examples is a pure product polynomial. The method uses algebraic number theory and combinatorial computations and relies on showing that a polynomial with bounded degree, restricted coefficients, and a zero of high order at 1 automatically vanishes at several roots of unity.