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Non-negative polynomials without hyperbolic certificates of non-negativity

H. L. Brian Ng, James Saunderson

Abstract

In this paper we study the relationship between the set of all non-negative multivariate homogeneous polynomials and those, which we call hyperwrons, whose non-negativity can be deduced from an identity involving the Wronskians of hyperbolic polynomials. We give a sufficient condition on positive integers $m$ and $2y$ such that there are non-negative polynomials of degree $2y$ in $m$ variables that are not hyperwrons. Furthermore, we give an explicit example of a non-negative quartic form that is not a sum of hyperwrons. We partially extend our results to hyperzouts, which are polynomials whose non-negativity can be deduced from an identity involving the Bézoutians of hyperbolic polynomials.

Non-negative polynomials without hyperbolic certificates of non-negativity

Abstract

In this paper we study the relationship between the set of all non-negative multivariate homogeneous polynomials and those, which we call hyperwrons, whose non-negativity can be deduced from an identity involving the Wronskians of hyperbolic polynomials. We give a sufficient condition on positive integers and such that there are non-negative polynomials of degree in variables that are not hyperwrons. Furthermore, we give an explicit example of a non-negative quartic form that is not a sum of hyperwrons. We partially extend our results to hyperzouts, which are polynomials whose non-negativity can be deduced from an identity involving the Bézoutians of hyperbolic polynomials.

Paper Structure

This paper contains 25 sections, 36 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Related Work
  4. Outline
  5. Preliminaries
  6. Semi-algebraic sets and functions
  7. Hyperbolic polynomials
  8. Relationship between non-negative polynomials and sums of squares
  9. Hyperbolic certificates of non-negativity
  10. Hyperbolic-Wronskian certificates
  11. Hyperbolic-Bézoutian certificates
  12. Relationship between hyperwrons and sums of squares
  13. Dimension analysis
  14. Preliminary facts about dimension of semi-algebraic sets
  15. Wronskian certificates
  16. ...and 10 more sections

Key Result

Theorem 1.1

If $m,y$ are positive integers such that then there exists a non-negative homogeneous polynomial in $m$ variables of degree $2y$ that is not a hyperwron.

Theorems & Definitions (75)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Bochnak Bochnak1998RealGeometry
  • Definition 2.2: Bochnak Bochnak1998RealGeometry
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 65 more