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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On a Problem of Byrnes concerning Polynomials with Restricted Coefficients


Author: David W. Boyd
Journal: Math. Comp. 66 (1997), 1697-1703
MSC (1991): Primary 11C08, 12D10; Secondary 94A99, 11Y99
MathSciNet review: 1433263
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a question of Byrnes concerning the minimal degree $n$ of a polynomial with all coefficients in $\{-1,1\}$ which has a zero of a given order $m$ at $x = 1$. For $m \le 5$, we prove his conjecture that the monic polynomial of this type of minimal degree is given by $\prod _{k=0}^{m-1} (x^{2^{k}}-1)$, but we disprove this for $m \ge 6$. We prove that a polynomial of this type must have $n \ge e^{\sqrt {m}(1 + o(1))}$, which is in sharp contrast with the situation when one allows coefficients in $\{-1,0,1\}$. The proofs use simple number theoretic ideas and depend ultimately on the fact that $-1 \equiv 1 \pmod 2$.


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Additional Information

David W. Boyd
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
Email: boyd@math.ubc.ca

DOI: http://dx.doi.org/10.1090/S0025-5718-97-00892-2
PII: S 0025-5718(97)00892-2
Keywords: Polynomial, zero, antenna array, notch filter
Received by editor(s): November 16, 1995
Additional Notes: This research was supported by a grant from NSERC
Article copyright: © Copyright 1997 American Mathematical Society