Approximating sums of squares with a single square

In this paper, we explore the following question. Given a trigonometric polynomial q(z1, … , zd) of several variables that is non-negative on the d-torus, how does one best approximate q with a (possibly outer) single modulus square? Our answer will lie in the notion of an outer component, which coincides with the outer factor in the case of one variable. The outer component may be computed numerically using semidefinite programming. We shall derive some properties of outer components, as well as pose some open problems.