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

Authors:	David W Boyd

Institution:	Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

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 at . 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 that $N^{}(6) = 48$ rather than . A polynomial with was exhibited for , 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 or . 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 th root of unity where $p \le m+1$ is a prime, then the condition that all coefficients of be in $\{-1,1\}$ , together with the requirement that be divisible by $(x-1)^{m}$ puts severe restrictions on the possible values for the cyclotomic integer $P(\zeta _{p})$ .

Keywords:	Polynomial zero spectral-null code

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