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

DOI:
https://doi.org/10.1090/S0025-5718-97-00892-2

MathSciNet review:
1433263

Full-text PDF Free Access

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

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