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

 

Littlewood polynomials with high order zeros


Authors: Daniel Berend and Shahar Golan
Journal: Math. Comp. 75 (2006), 1541-1552
MSC (2000): Primary 11B83, 12D10; Secondary 94B05, 11Y99
Published electronically: May 1, 2006
MathSciNet review: 2219044
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Abstract: Let $ N^{*}(m)$ be the minimal length of a polynomial with $ \pm1$ coefficients divisible by $ (x-1)^m$. Byrnes noted that $ N^{*}(m)\leq2^m$ for each $ m$, and asked whether in fact $ N^{*}(m)=2^m$. Boyd showed that $ N^{*}(m) = 2^{m}$ for all $ m \le 5$, but $ N^{*}(6) = 48$. He further showed that $ N^*(7)=96$, and that $ N^{*}(8)$ is one of the 5 numbers $ 96, 144, 160, 176$, or $ 192$. Here we prove that $ N^{*}(8) = 144$. Similarly, let $ m^*(N)$ be the maximal power of $ (x-1)$ dividing some polynomial of degree $ N-1$ with $ \pm1$ coefficients. Boyd was able to find $ m^*(N)$ for $ N<88$. In this paper we determine $ m^*(N)$ for $ N<168$.


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Additional Information

Daniel Berend
Affiliation: Department of Computer Science, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105 Israel
Email: berend@cs.bgu.ac.il

Shahar Golan
Affiliation: Department of Computer Science, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105 Israel
Email: golansha@cs.bgu.ac.il

DOI: http://dx.doi.org/10.1090/S0025-5718-06-01848-5
PII: S 0025-5718(06)01848-5
Keywords: Littlewood polynomials, spectral-null code, antenna array
Received by editor(s): May 5, 2005
Received by editor(s) in revised form: June 30, 2005
Published electronically: May 1, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.