The average norm of polynomials of fixed height

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

$\beta_n(m):=\frac{1}{3^{n+1}}\sum_{P\in \mathfrak{F}_n}\Vert P\Vert _m^m.$

Here $\Vert P\Vert _m^m$ 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

$\beta_n(2)=\frac23 (n+1),$
$\beta_n(4)=\frac89 n^2+\frac{14}{9}n+\frac23$

and

$\beta_n(6)=\frac{16}{9}n^3+4n^2+\frac{26}{9}n+\frac23.$