The average norm of polynomials of fixed height

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 \[ \beta _n(6)=\frac {16}{9}n^3+4n^2+\frac {26}{9}n+\frac 23. \]