An optimization problem arising in CR geometry

Abstract

We determine the asymptotic behavior as the degree tends to infinity of the minimal L1 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→m(n,d) is monotone increasing in d. We prove that limd→∞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 limd→∞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.