Average growth of Lp norms of Erdős–Szekeres polynomials

We study the average growth of pth powers of Lp noms on the unit circle of Erdős–Szekeres polynomials Pn({sj},z)=∏j=1n(1-zsj)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{n}( \{ s_{j}\} ,z) = \prod_{j=1}^{n}(1-z^{s_{j}})$$\end{document} where 1≤s1,s2,…,sn≤M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \leq s_{1},s_{2} , \ldots ,s_{n} \leq M$$\end{document} and M,n→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M,n\rightarrow \infty $$\end{document}. In particular, we show the average growth is geometric and determine the precise geometric growth. We also analyze the variance.