Polynomials with height 1 and prescribed vanishing at 1

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 $m\leq10$ 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 $m\leq21$. 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.