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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

As in the earlier paper with this title, we consider a question of Byrnes concerning the minimal length of a polynomial with all coefficients in which has a zero of a given order at . In that paper we showed that for all and showed that the extremal polynomials for were those conjectured by Byrnes, but for that 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 is one of the 7 values or . Here we prove that without determining all extremal polynomials. We also make some progress toward determining . As in the previous paper, we use a combination of number theoretic ideas and combinatorial computation. The main point is that if is a primitive th root of unity where is a prime, then the condition that all coefficients of be in , together with the requirement that be divisible by puts severe restrictions on the possible values for the cyclotomic integer .

**[BM]**P. Borwein and M. Mossinghoff,*Polynomials with height**and prescribed vanishing at*, Experiment. Math.**9**(2000), 425-433. CMP**2001:04****[Bo]**David W. Boyd,*On a problem of Byrnes concerning polynomials with restricted coefficients*, Math. Comp.**66**(1997), no. 220, 1697–1703. MR**1433263**, 10.1090/S0025-5718-97-00892-2**[By]**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 and J.F. Byrnes, eds.), Kluwer Academic Publishers, Dordrecht, 1990, pp. 677-678.**[E]**Harold M. Edwards,*Fermat’s last theorem*, Graduate Texts in Mathematics, vol. 50, Springer-Verlag, New York-Berlin, 1977. A genetic introduction to algebraic number theory. MR**616635****[FL]**Gregory Freiman and Simon Litsyn,*Asymptotically exact bounds on the size of high-order spectral-null codes*, IEEE Trans. Inform. Theory**45**(1999), no. 6, 1798–1807. MR**1720633**, 10.1109/18.782100**[LN]**Rudolf Lidl and Harald Niederreiter,*Finite fields*, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR**746963****[NW]**Albert Nijenhuis and Herbert S. Wilf,*Combinatorial algorithms*, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. For computers and calculators; Computer Science and Applied Mathematics. MR**510047****[RSV]**R.M. Roth, P.H. Siegel, and A. Vardy,*High-order spectral-null codes: Constructions and bounds*, IEEE Trans. Inform. Theory**35**(1989), 463-472.**[R]**Ron M. Roth,*Spectral-null codes and null spaces of Hadamard submatrices*, Des. Codes Cryptogr.**9**(1996), no. 2, 177–191. MR**1409444**, 10.1007/BF00124592**[S]**V. Skachek,*Coding for Spectral-Null Constraints*, Research Thesis, Master of Science in Computer Science, Technion, Israel Institute of Technology, November 1997 (Hebrew).

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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/S0025-5718-01-01360-6

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