From the text: “A polynomial \(f \in \mathbb C[z]\) is unimodular if all its coefficients have unit modulus. Let \(U_n\) denote the set of unimodular polynomials of degree \(n-1\), and let \(U_n^\ast\) denote the subset of reciprocal unimodular polynomials, which have the property that \(f(z) = \omega z^{n-1} \overline{f(1/\overline{z})}\) for some complex number \(\omega\) with \(|\omega|=1\). We study the geometric and arithmetic mean values of both the normalized Mahler’s measure \(M(f)/\sqrt{n}\) and \(L_p\) norm \(\|f\|_p/\sqrt{n}\) over the sets \(U_n\) and \(U_n^\ast\), and compute asymptotic values in each case.
Recall that Mahler’s measure of a polynomial \(f(z)\in\mathbb C(z)\), denoted by \(M(f)\), is defined by \[ \log M(f)= \int^1_0 \log|f(e^{2\pi it})| dt. \] If \(f(z)= a_{n-1}\prod_{k=1}^{n-1} (z- \alpha_k)\) then, from Jensen’s formula, we have \(M(f)= |a_{n-1}|\cdot \prod_{k=1}^{n-1}\max\{1,|\alpha_k|\}\). Mathematicians like Littlewood and Kahame worked on problems around these concepts.
We show for example that both the geometric and arithmetic mean of the normalized Mahler’s measure approach \(e^{-\gamma/2} = 0.749306\dots\) as \(n\rightarrow \infty\) for unimodular polynomials, and \(e^{-\gamma/2}/\sqrt{2} = 0.529839\dots\) for reciprocal unimodular polynomials. We also show that for large \(n\), almost all polynomials in these sets have normalized Mahler’s measure or \(L_p\) norm very close to the respective limiting mean value.”