On a Waring's problem for Hermitian lattices

Abstract

Assume E is an imaginary quadratic field and O is its ring of integers. For each positive integer m, let Im be the free Hermitian lattice of rank m over O having an orthonormal basis. For each positive integer n, let SO(n) be the set of all Hermitian lattices of rank n over O that can be represented by some Im. Denote by gO(n) the smallest positive integer g such that each Hermitian lattice in SO(n) can be represented by Ig. In this paper, we shall provide an explicit upper bound for gO(n) for all imaginary quadratic fields E and all positive integers n.