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

Author: David W. Boyd
Journal: Math. Comp. 71 (2002), 1205-1217
MSC (2000): Primary 11C08, 12D10; Secondary 94B05, 11Y99
Published electronically: May 11, 2001
MathSciNet review: 1898751
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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 $P(\zeta _{p})$.

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Additional Information

David W. Boyd
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Keywords: Polynomial, zero, spectral-null code
Received by editor(s): September 8, 1997
Received by editor(s) in revised form: September 19, 2000
Published electronically: May 11, 2001
Additional Notes: This research was supported by a grant from NSERC
Article copyright: © Copyright 2001 American Mathematical Society