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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Abstract | References | Similar Articles | Additional Information

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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Additional Information

**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.

Article copyright:
© Copyright 2006
by the authors