Littlewood polynomials with high order zeros

Abstract:

Let $N^{*}(m)$ be the minimal length of a polynomial with $\pm 1$ coefficients divisible by $(x-1)^m$. Byrnes noted that $N^{*}(m)\leq 2^m$ for each $m$, and asked whether in fact $N^{*}(m)=2^m$. Boyd showed that $N^{*}(m) = 2^{m}$ for all $m \le 5$, but $N^{*}(6) = 48$. He further showed that $N^*(7)=96$, and that $N^{*}(8)$ is one of the 5 numbers $96, 144, 160, 176$, or $192$. Here we prove that $N^{*}(8) = 144$. Similarly, let $m^*(N)$ be the maximal power of $(x-1)$ dividing some polynomial of degree $N-1$ with $\pm 1$ coefficients. Boyd was able to find $m^*(N)$ for $N<88$. In this paper we determine $m^*(N)$ for $N<168$.