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# Norms of cyclotomic Littlewood polynomials

Published online by Cambridge University Press:  24 February 2005

## 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