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Norms of cyclotomic Littlewood polynomials

Published online by Cambridge University Press:  24 February 2005

PETER BORWEIN
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
KWOK-KWONG STEPHEN CHOI
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
RON FERGUSON
Affiliation:
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6 Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2

Abstract

The main result of this paper gives an explicit computation of the $L_4$ norm of any cyclotomic polynomial of the form \[ f(x)=\Phi_{p_1}(\pm x)\Phi_{p_2}(\pm x^{p_1})\cdots \Phi_{p_r}(\pm x^{p_1 p_2\cdots p_{r-1}}). \] Here $\Phi_{p}$ is the $p$th cyclotomic polynomial and the $p_i$ are primes that are not necessarily distinct.

A corollary of this is the following theorem.

THEOREM. If\[ f(x)=\Phi_{p_1}(\e_1 x)\Phi_{p_2}(\e_2 x^{p_1})\cdots \Phi_{p_r}(\e_r x^{p_1 p_2\cdots p_{r-1}}), \]where N = p1p2···prand εj = ±, then \[ \frac{\| f\|_4^4}{N^2} & \ge & \frac{\big\|\Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big) \big\|_4^4}{4^r}\\& = & \frac{\big(\frac12 +\frac{5}{34}\sqrt{17}\big)(1+\sqrt{17})^r - \big({-}\frac12 +\frac{5}{34}\sqrt{17}\big)(1-\sqrt{17})^r}{4^r}. \] In particular the minimum possible normalized L4 norm of any polynomial of the above form is attained by \[ \Phi_2(-x)\Phi_2(-x^2)\cdots \Phi_2\big({-}x^{2^{r-1}}\big). \]

Type
Research Article
Copyright
2005 Cambridge Philosophical Society

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