Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Remote Access
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

An extremal property of Fekete polynomials


Authors: Peter Borwein, Kwok-Kwong Stephen Choi and Soroosh Yazdani
Journal: Proc. Amer. Math. Soc. 129 (2001), 19-27
MSC (2000): Primary 11J54, 11B83
Published electronically: July 21, 2000
MathSciNet review: 1784013
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

The Fekete polynomials are defined as


\begin{displaymath}F_q(z) := \sum^{q-1}_{k=1} \left(\frac{k}{q}\right) z^k\end{displaymath}

where $\left(\frac{\cdot}{q}\right)$ is the Legendre symbol. These polynomials arise in a number of contexts in analysis and number theory. For example, after cyclic permutation they provide sequences with smallest known $L_4$ norm out of the polynomials with $\pm 1$ coefficients.

The main purpose of this paper is to prove the following extremal property that characterizes the Fekete polynomials by their size at roots of unity.



Theorem 0.1. Let $f(x)=a_1x+a_2x^2+\cdots +a_{N-1}x^{N-1}$ with odd $N$ and $a_n=\pm 1$. If

\begin{displaymath}\operatorname{max}\{ \vert f(\omega^k)\vert : 0 \le k \le N-1 \} = \sqrt{N}, \end{displaymath}

then $N$ must be an odd prime and $f(x)$ is $\pm F_q(x)$. Here $ \omega:=e^{\frac{2\pi i}{N}}.$



This result also gives a partial answer to a problem of Harvey Cohn on character sums.


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

  • 1. Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR 0434929 (55 #7892)
  • 2. Leonard D. Baumert, Cyclic difference sets, Lecture Notes in Mathematics, Vol. 182, Springer-Verlag, Berlin-New York, 1971. MR 0282863 (44 #97)
  • 3. A. Biró, Notes on a Problem of H. Cohn, J. Number Theory, 77 (1999), 200-208. CMP 99:16
  • 4. P. Borwein and K-K. Choi, Explicit Merit Factor Formulae For Fekete and Turyn Polynomials, (in press).
  • 5. K-K. Choi and M-K Siu, Counter-Examples to a Problem of Cohn on Classifying Characters, J. Number Theory, to appear.
  • 6. B. Conrey, A.Granville and B.Poonen, Zeros of Fekete Polynomials, (in press).
  • 7. S-L Ma, M-K Siu and Z Zheng, On a Problem of Cohn on Character Sums, (in press).
  • 8. H.L. Montgomery, An Exponential Sum Formed with the Legendre Symbol, Acta Arith, 37 (1980), 375-380
  • 9. Hugh L. Montgomery, Ten lectures on the interface between analytic number theory and harmonic analysis, CBMS Regional Conference Series in Mathematics, vol. 84, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1297543 (96i:11002)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 11J54, 11B83

Retrieve articles in all journals with MSC (2000): 11J54, 11B83


Additional Information

Peter Borwein
Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: pborwein@math.sfu.ca

Kwok-Kwong Stephen Choi
Affiliation: Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, SAR, China
Email: choi@maths.hku.hk

Soroosh Yazdani
Affiliation: Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: syazdani@undergrad.math.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05798-1
PII: S 0002-9939(00)05798-1
Keywords: Class number, $\pm1$ coefficients, merit factor, Fekete polynomials, Turyn polynomials, Littlewood polynomials
Received by editor(s): March 15, 1999
Published electronically: July 21, 2000
Additional Notes: The research of P. Borwein is supported, in part, by NSERC of Canada. K.K. Choi is a Pacific Institute of Mathematics Postdoctoral Fellow and the Institute’s support is gratefully acknowledged.
Communicated by: Dennis A. Hejhal
Article copyright: © Copyright 2000 Copyright held by the authors



Comments: Email Webmaster

© Copyright , American Mathematical Society
Contact Us · Sitemap · Privacy Statement

Connect with us Facebook Twitter Google+ LinkedIn Instagram RSS feeds Blogs YouTube Podcasts Wikipedia