On the supremum of a quotient of power sums

TL;DR

This paper investigates the asymptotic behavior of a quotient of power sums for real vectors, deriving conditions for polynomial positivity constraints in matrices, with applications in mathematical finance and price impact models.

Contribution

It introduces a new homogeneous quotient involving power sums and establishes its linear growth with dimension, linking it to matrix positivity conditions.

Findings

Supremum of the quotient grows linearly with dimension

Derived conditions for polynomial positivity in matrices

Applied results to price impact models in finance

Abstract

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.