Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

SDP Feasibility Problems and sos Representation Ranks for OT-FKM Type Isoparametric Polynomials

Jianquan Ge, Kai Jia, Yuyang Zhao

Abstract

Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study the quartic forms GF = (|x|^4 + F(x))/2 associated with isoparametric polynomials F of OT-FKM type with g = 4. We characterize the sos property of GF in terms of the feasibility of an explicit SDP determined by the underlying Clifford system, and in the sos cases we obtain quantitative rank bounds for sos representations, with rigidity when m >= 3.

SDP Feasibility Problems and sos Representation Ranks for OT-FKM Type Isoparametric Polynomials

Abstract

Semidefinite programming (SDP) provides a fundamental framework for studying properties of sum-of-squares (sos) representations of nonnegative polynomials. In this paper we study the quartic forms GF = (|x|^4 + F(x))/2 associated with isoparametric polynomials F of OT-FKM type with g = 4. We characterize the sos property of GF in terms of the feasibility of an explicit SDP determined by the underlying Clifford system, and in the sos cases we obtain quantitative rank bounds for sos representations, with rigidity when m >= 3.
Paper Structure (18 sections, 26 theorems, 233 equations, 2 tables)

This paper contains 18 sections, 26 theorems, 233 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. SDP Characterization for the sos Property of $G_F$
  4. A Reduction to Representative Cases for Theorem \ref{['sos thm']}
  5. The Non-sos Cases in Theorem \ref{['sos thm']}
  6. The Non-sos Case $(m_+,m_-)=(4,3)^D$
  7. The Non-sos Cases $(m,l)=(3,4r)$ with $r\ge 3$
  8. The sos Cases in Theorem \ref{['sos thm']}
  9. Constructing Feasible Matrices for $(m,l)=(1,k+2)$
  10. Constructing Feasible Matrices for $(m,l)=(2,2k+2)$
  11. The Unique Feasible Matrix for $(m,l)=(6,8)$
  12. Ranks of sos Representations and the Proof of Theorem \ref{['rank thm']}
  13. Ranks of sos Representations via SDP
  14. Rank Sets of sos Representations of $G_F$
  15. Possible Ranks for $(m_+,m_-)=(3,4),(4,3)^I,(5,2),(6,1)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G_F$ be the psd form in nonnegativepolyG on $\mathbb{R}^n$ associated with an OT-FKM type isoparametric polynomial $F$. Then $G_F$ is sos if and only if the following SDP feasibility problem admits a solution in the matrix $B$: where $B=(B_{ik})_{i,k=1}^{l}$ is viewed as an $l\times l$ block matrix with blocks $B_{ik}\in\mathbb{R}^{l\times l}$, and $R_i$ is defined in Define R_q from the Cli

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 39 more