On a problem of Byrnes concerning polynomials with restricted coefficients, II

Abstract:

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length $N^{*}(m)$ of a polynomial with all coefficients in $\{-1,1\}$ which has a zero of a given order $m$ at $x = 1$. In that paper we showed that $N^{*}(m) = 2^{m}$ for all $m \le 5$ and showed that the extremal polynomials for were those conjectured by Byrnes, but for $m = 6$ that $N^{*}(6) = 48$ rather than $64$. A polynomial with $N = 48$ was exhibited for $m = 6$, but it was not shown there that this extremal was unique. Here we show that the extremal is unique. In the previous paper, we showed that $N^{*}(7)$ is one of the 7 values $48, 56, 64, 72, 80, 88$ or $96$. Here we prove that $N^{*}(7) = 96$ without determining all extremal polynomials. We also make some progress toward determining $N^{*}(8)$. As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if $\zeta _{p}$ is a primitive $p$th root of unity where $p \le m+1$ is a prime, then the condition that all coefficients of $P$ be in $\{-1,1\}$, together with the requirement that $P(x)$ be divisible by $(x-1)^{m}$ puts severe restrictions on the possible values for the cyclotomic integer $P(\zeta _{p})$.