On products of polynomials

Abstract:

For a complex polynomial $P$ in a single variable, let $H(P)$ be the maximum of the absolute values of its coefficients. Given nonnegative integers ${n_1}$ and ${n_2}$, it is well known that $\mu ({n_1},{n_2}) = \inf H({P_1}{P_2})/(H({P_1})H({P_2})) > 0$, where the infimum is taken over all such polynomials ${P_1}$ and ${P_2}$ of degrees ${n_1}$ and ${n_2}$ respectively. We determine here the exact values of $\mu (1,n)$ for every $n$.