Remote Access Mathematics of Computation
Green Open Access

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
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 $-1 \equiv 1 \pmod 2$.

References [Enhancements On Off] (What's this?)

  • Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR 0434929
  • P. Borwein, T. Erdélyi & G. Kós, Polynomials with Restricted Coefficients (to appear).
  • J.S. Byrnes & D.J. Newman, Null Steering Employing Polynomials with Restricted Coefficients, IEEE Trans. Antennas and Propagation 36 (1988), 301–303.
  • J.S. Byrnes, Problems on Polynomials with Restricted Coefficients Arising from Questions in Antenna Array Theory, Recent Advances in Fourier Analysis and Its Applications (J.S. Byrnes & J.F. Byrnes, eds.), Kluwer Academic Publishers, Dordrecht, 1990, pp. 677–678.
  • M. Mignotte, Sur les polynômes divisibles par $(X-1)^{n}$, Arithmetix 2 (1980), 28–29.
  • Aleksej D. Korshunov, On the number of nonisomorphic strongly connected finite automata, Elektron. Informationsverarb. Kybernet. 22 (1986), no. 9, 459–462 (English, with German and Russian summaries). MR 862029
  • J. Barkley Rosser and Lowell Schoenfeld, Approximate formulas for some functions of prime numbers, Illinois J. Math. 6 (1962), 64–94. MR 137689

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (1991): 11C08, 12D10, 94A99, 11Y99

Retrieve articles in all journals with MSC (1991): 11C08, 12D10, 94A99, 11Y99

Additional Information

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

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