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On the degree of varieties of sum of squares

Andrew Ferguson, Giorgio Ottaviani, Mohab Safey El Din, Ettore Teixeira Turatti

TL;DR

It is shown that SOS 2 ( f ) is isomorphic to O(2) and hence the degree bound becomes an equality and the dimension of SOS k ( f) is computed and its degree is bounded from below by the degree of O( k ) .

Abstract

We study the problem of how many different sums of squares decompositions a general polynomial $f$ with SOS-rank $k$ admits. We show that there is a link between the variety $\mathrm{SOS}_k(f)$ of all SOS-decompositions of $f$ and the orthogonal group $\mathrm{O}(k)$. We exploit this connection to obtain the dimension of $\mathrm{SOS}_k(f)$ and show that its degree is bounded from below by the degree of $\mathrm{O}(k)$. In particular, for $k=2$ we show that $\mathrm{SOS}_2(f)$ is isomorphic to $\mathrm{O}(2)$ and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank $k$ and obtain the degree in the special case $k=2$.

On the degree of varieties of sum of squares

TL;DR

It is shown that SOS 2 ( f ) is isomorphic to O(2) and hence the degree bound becomes an equality and the dimension of SOS k ( f) is computed and its degree is bounded from below by the degree of O( k ) .

Abstract

We study the problem of how many different sums of squares decompositions a general polynomial with SOS-rank admits. We show that there is a link between the variety of all SOS-decompositions of and the orthogonal group . We exploit this connection to obtain the dimension of and show that its degree is bounded from below by the degree of . In particular, for we show that is isomorphic to and hence the degree bound becomes an equality. Moreover, we compute the dimension of the space of polynomials of SOS-rank and obtain the degree in the special case .
Paper Structure (8 sections, 17 theorems, 62 equations, 1 table)

This paper contains 8 sections, 17 theorems, 62 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Prior works
  4. Main results
  5. Structure of the paper
  6. Preliminaries
  7. The degree of the variety of all SOS decompositions
  8. The degree of the variety of the sum of two squares

Key Result

Theorem 1.4

Let $f\in\mathrm{SOS}_2\subset\mathrm{Sym}^{2d}V$, $\dim V=n+1>2$, be a generic polynomial that is the sum of two squares. Then, $\mathrm{SOS}_2(f)$ has two irreducible components isomorphic to $\mathrm{SO}(2)$.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 28 more