## The average norm of polynomials of fixed height

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- by Peter Borwein and Kwok-Kwong Stephen Choi PDF
- Trans. Amer. Math. Soc.
**359**(2007), 923-936

## Abstract:

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
- 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.
- © Copyright 2006 by the authors
- 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
- MathSciNet review: 2255202