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 .

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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:
https://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