On the supremum of a quotient of power sums

We define a function of two real vectors by a certain homogeneous quotient involving power sums, and show that its supremum grows asymptotically linearly w.r.t. the dimension. From this, we deduce a condition under which a parametric set of real matrices satisfies a set of polynomial positivity constraints. This characterization finds an application in mathematical finance, in a recent study on price impact models.