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Biquadratic SOS Rank: Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

Chunfeng Cui, Liqun Qi, Yi Xu

TL;DR

The paper investigates sum-of-squares decompositions for biquadratic forms of bidegree $(2,2)$ and introduces the SOS rank invariant $ ext{BSR}(m,n)$. It proves exact values $ ext{BSR}(3,3)=6$ and $ ext{BSR}(4,3)=7$, consistent with a conjectured linear pattern $ ext{BSR}(m,n)=m+n$ for $m,n\nge3$, and it derives a general upper bound $ ext{BSR}(m,n)\le mn-2$ for all $(m,n) eq(2,2)$ by combining projective normality of Segre-Veronese embeddings with Sard’s theorem and a Rank-Theorem-based analysis. The work develops a systematic framework that extends to general dimensions, showing that every $m imes n$ SOS biquadratic form is a sum of at most $mn-2$ squares, while highlighting a substantial gap between this upper bound and the known lower bound $m+n$. These results influence PSD vs SOS distinctions in higher dimensions and provide a pathway toward a complete formula for $ ext{BSR}(m,n)$ in broader settings.

Abstract

We prove that every $3 \times 3$ sum-of-squares (SOS) biquadratic form can be expressed as the sum of at most \textbf{six} squares of bilinear forms, establishing $\mathrm{BSR}(3,3) = 6$. We also determine the exact SOS rank for $4 \times 3$ biquadratic forms: $\mathrm{BSR}(4,3)=7$. These results fit the pattern $\mathrm{BSR}(m,n)=m+n$, leading to the conjecture that this linear formula holds for all $m,n \ge 3$. Furthermore, we extend our geometric-analytic method to general dimensions and show that for any integers $m,n \ge 2$ with $(m,n)\neq(2,2)$, every $m \times n$ SOS biquadratic form is a sum of at most $mn-2$ squares, improving the general upper bound of $mn-1$ established in earlier work. For the $3 \times 3$ case, we provide a complete geometric analysis of the SOS cone structure, and for general dimensions we establish a systematic framework that applies to all $m \times n$ biquadratic forms except the degenerate $(2,2)$ case. We note that the lower bound of 6 for $3 \times 3$ forms is achieved by a simple biquadratic form, and for general $m,n\ge 3$, it is known that the maximum SOS rank is at least $m+n$. Our results establish new upper bounds and significantly reduce the gap between the lower and upper bounds for the worst-case SOS rank of biquadratic forms across all dimensions.

Biquadratic SOS Rank: Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

TL;DR

The paper investigates sum-of-squares decompositions for biquadratic forms of bidegree and introduces the SOS rank invariant . It proves exact values and , consistent with a conjectured linear pattern for , and it derives a general upper bound for all by combining projective normality of Segre-Veronese embeddings with Sard’s theorem and a Rank-Theorem-based analysis. The work develops a systematic framework that extends to general dimensions, showing that every SOS biquadratic form is a sum of at most squares, while highlighting a substantial gap between this upper bound and the known lower bound . These results influence PSD vs SOS distinctions in higher dimensions and provide a pathway toward a complete formula for in broader settings.

Abstract

We prove that every sum-of-squares (SOS) biquadratic form can be expressed as the sum of at most \textbf{six} squares of bilinear forms, establishing . We also determine the exact SOS rank for biquadratic forms: . These results fit the pattern , leading to the conjecture that this linear formula holds for all . Furthermore, we extend our geometric-analytic method to general dimensions and show that for any integers with , every SOS biquadratic form is a sum of at most squares, improving the general upper bound of established in earlier work. For the case, we provide a complete geometric analysis of the SOS cone structure, and for general dimensions we establish a systematic framework that applies to all biquadratic forms except the degenerate case. We note that the lower bound of 6 for forms is achieved by a simple biquadratic form, and for general , it is known that the maximum SOS rank is at least . Our results establish new upper bounds and significantly reduce the gap between the lower and upper bounds for the worst-case SOS rank of biquadratic forms across all dimensions.
Paper Structure (45 sections, 30 theorems, 41 equations, 1 table)

This paper contains 45 sections, 30 theorems, 41 equations, 1 table.

Table of Contents

  1. Introduction
  2. Historical Context
  3. Biquadratic Forms and SOS Rank
  4. The PSD vs. SOS Distinction and BSR
  5. Previous Bounds and Open Problems
  6. Our Contributions
  7. Structure of the Paper
  8. Preliminaries
  9. Spaces of Bilinear and Biquadratic Forms
  10. Positive Semidefiniteness and Sum of Squares
  11. The SOS Map and Its Derivative
  12. Stratification by Common Zeros
  13. Projective Normality of Segre-Veronese Embeddings
  14. A Reduction Lemma for Point Evaluations
  15. Tools from Analysis: Rank Theorem and Sard's Theorem
  16. ...and 30 more sections

Key Result

Theorem 2.4

The Segre-Veronese embedding by $\mathcal{O}(2,2)$ is projectively normal for all $m, n \ge 2$hartshorneSt96.

Theorems & Definitions (58)

  • Example 2.1
  • Example 2.2
  • Definition 2.3: Biquadratic SOS Rank
  • Theorem 2.4: Projective Normality of Segre-Veronese
  • Lemma 2.5: Reduction of point evaluations
  • proof
  • Theorem 2.6: Rank Theorem
  • Theorem 2.7: Sard's Theorem
  • Example 3.1
  • Lemma 3.2: Closedness of the image
  • ...and 48 more