An optimization problem arising in CR geometry

Abstract

We determine the asymptotic behavior as the degree tends to infinity of the minimal $L^1$ norm $m(n,d)$ of the solution of an optimization problem arising when studying polynomial sphere maps. Here n is the source dimension and d is the degree. We provide upper and lower bounds for $m(n,d)$. We use these bounds to show that the function $d\to m(n,d)$ is monotone increasing in d. We prove that ${lim}_{d\to \mathrm{\infty }}\frac{m(n,d)}{d}=n(n-1)$. Let $N(n,d)$ denote the minimum possible target dimension of a monomial sphere map of degree d. We show, in source dimension unequal to 2, that ${lim}_{d\to \mathrm{\infty }}\frac{m(n,d)}{N(n,d)}=n$. The limit is 4 when $n=2$. We discuss some complicated results obtained by coding when $n=2$.