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On the supremum of a quotient of power sums

Stefan Gerhold, Friedrich Hubalek

TL;DR

The paper analyzes a homogeneous quotient of power sums $Q(x,y)$ and proves that its supremum grows linearly with dimension, establishing a sharp constant $c^*=(7\sqrt{7}-17)/27$ governing $Q(x,y)\!<\!c^*n$ and its asymptotic limit. It combines a structural reduction to low-dimensional configurations with a one-dimensional optimization to identify the maximizing parameters, and constructs explicit extremal sequences to prove sharpness. The results yield a concrete condition for a parametric family of real matrices to satisfy polynomial positivity constraints, with applications to price-impact models in mathematical finance. The work also provides detailed numerical bounds for specific matrix families and discusses large-dimension behavior.

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.

On the supremum of a quotient of power sums

TL;DR

The paper analyzes a homogeneous quotient of power sums and proves that its supremum grows linearly with dimension, establishing a sharp constant governing and its asymptotic limit. It combines a structural reduction to low-dimensional configurations with a one-dimensional optimization to identify the maximizing parameters, and constructs explicit extremal sequences to prove sharpness. The results yield a concrete condition for a parametric family of real matrices to satisfy polynomial positivity constraints, with applications to price-impact models in mathematical finance. The work also provides detailed numerical bounds for specific matrix families and discusses large-dimension behavior.

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.

Paper Structure

This paper contains 4 sections, 5 theorems, 57 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Application
  4. Numerical values

Key Result

Theorem 2.1

With we have and Both assertions remain true if the condition $x\in\mathbb{R}^n_{>0}$ is replaced by $x\in\mathbb{R}^{n+1}_{>0}$.

Theorems & Definitions (10)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:main']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof