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On nonnegative invariant quartics in type A

Sebastian Debus, Charu Goel, Salma Kuhlmann, Cordian Riener

Abstract

The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse $A_n$-invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.

On nonnegative invariant quartics in type A

Abstract

The equivariant nonnegativity versus sums of squares question has been solved for any infinite series of essential reflection groups but type A. As a first step to a classification, we analyse -invariant quartics. We prove that the cones of invariant sums of squares and nonnegative forms are equal if and only if the number of variables is at most 3 or odd.
Paper Structure (9 sections, 13 theorems, 29 equations)

This paper contains 9 sections, 13 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. The reflection groups of type $A$ and $A_n$-invariant polynomials
  3. SOS versus PSD for $A_n$-invariant Quartics
  4. Global nonnegativity versus nonnegativity on $U_n$
  5. PSD $A_n$ invariant quartics
  6. Verification of the extremality of $G_n$
  7. Verification of the extremality of $F_n$
  8. SOS $A_n$-invariant quartics
  9. Proof of Theorem \ref{['thm:main']}

Key Result

Proposition 2.1

The ring homomorphism $\mathbb{R}[\mathbf{y}] \to \mathbb{R}[\mathbf{x}]/(\mathbf{x}_1+\ldots+\mathbf{x}_{n+1})$ defined by $\mathbf{y}_i \mapsto \mathbf{x}_1 - \mathbf{x}_{i+1}$, for all $1 \leq i \leq n$, is a $A_n$-equivariant isomorphism.

Theorems & Definitions (23)

  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Definition 3.1
  • Theorem 3.2
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof : Proof of Lemma \ref{['lem:extr A psd']}
  • ...and 13 more