   ISSN 1088-6850(online) ISSN 0002-9947(print)

The average norm of polynomials of fixed height

Authors: Peter Borwein and Kwok-Kwong Stephen Choi
Journal: Trans. Amer. Math. Soc. 359 (2007), 923-936
MSC (2000): Primary 11C08, 26C05
DOI: https://doi.org/10.1090/S0002-9947-06-03952-3
Published electronically: September 12, 2006
MathSciNet review: 2255202
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Let $n\ge 0$ be any integer and let $\mathfrak {F}_n:=\left \{ \sum _{i=0}^na_iz^i : a_i = 0, \pm 1 \right \}$ be the set of all polynomials of height 1 and degree $n$. Let $\beta _n(m):=\frac {1}{3^{n+1}}\sum _{P\in \mathfrak {F}_n}\| P\|_m^m.$ Here $\| P\|_m^m$ is the $mth$ power of the $L_m$ norm on the boundary of the unit disc. So $\beta _n(m)$ is the average of the $mth$ power of the $L_m$ norm over $\mathfrak {F}_n.$ In this paper we give exact formulae for $\beta _n(m)$ for various values of $m$. We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients $\pm 1$ and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.

Theorem 0.1. For $n\ge 0$, we have \begin{gather*} \beta _n(2)=\frac 23 (n+1), \beta _n(4)=\frac 89 n^2+\frac {14}{9}n+\frac 23 \end{gather*} and $\beta _n(6)=\frac {16}{9}n^3+4n^2+\frac {26}{9}n+\frac 23.$

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

Kwok-Kwong Stephen Choi
Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
Email: kkchoi@cecm.sfu.ca

Keywords: Polynomials of height 1, Littlewood polynomials, average $L_p$ norm
Received by editor(s): June 19, 2001
Received by editor(s) in revised form: January 22, 2005
Published electronically: September 12, 2006
Additional Notes: The research of the first author was supported by MITACS and by NSERC of Canada, and the research of the second author was supported by NSERC of Canada.