On the SOS Rank of Simple and Diagonal Biquadratic Forms

We study the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms. For simple biquadratic forms in $3 \times 3$ variables, we show that the maximum SOS rank is exactly $6$, attained by a specific six-term form. We further prove that for any $m \ge 3$, there exists an $m \times m$ simple biquadratic form whose SOS rank is exactly $2m$. Moreover, we show that for all $m, n \ge 3$, the maximum SOS rank over $m \times n$ simple biquadratic forms is at least $m+n$, which implies $\mathrm{BSR}(m,n) \ge m+n$. For diagonal biquadratic forms with nonnegative coefficients, we prove an SOS rank upper bound of $7$, improving the general bound of $8$ for $3 \times 3$ forms. These results provide new lower and upper bounds on the worst-case SOS rank of biquadratic forms and highlight the role of structure in reducing the required number of squares.

💡 Research Summary

The paper investigates the sum‑of‑squares (SOS) rank of two natural subclasses of real biquadratic forms P(x,y)=∑{i,k=1}^m∑{j,l=1}^n a_{ijkl}x_i x_k y_j y_l: (i) simple forms, which contain only distinct monomials of the type x_i^2 y_j^2, and (ii) diagonal forms, which are of the form ∑{i=1}^m∑{j=1}^n a_{ij}x_i^2 y_j^2 with non‑negative coefficients. While any positive‑semidefinite (PSD) form is automatically SOS, the minimal number of squares required—the SOS rank—is highly non‑trivial.

3×3 simple forms. The authors exhibit the six‑term form
P′(x,y)=x_1^2y_1^2+x_2^2y_2^2+x_3^2y_3^2+x_1^2y_2^2+x_2^2y_3^2+x_3^2y_1^2,
which clearly decomposes into six bilinear squares, giving an upper bound of 6. To prove that five squares are impossible, they write a generic SOS representation P′=∑{t=1}^R L_t(x,y)^2 with L_t(x,y)=∑{i,j}c_{ij}^{(t)}x_i y_j. Comparing coefficients yields orthonormal vectors C_{ij}∈ℝ^R for each monomial x_i^2y_j^2, and orthogonality between distinct C_{ij}. Six mutually orthogonal unit vectors cannot live in ℝ^5, forcing R≥6. Hence sos(P′)=6, establishing that the maximum SOS rank for 3×3 simple biquadratics is exactly 6 (Theorem 2.5).

General m×m simple forms. For any m≥3 the paper defines
P_{m,m,2m}=∑{i=1}^m x_i^2 y_i^2+∑{i=1}^m x_i^2 y_{i+1}^2 (indices modulo m).
Each term is a single square, so sos≤2m. Using the same orthogonal‑vector argument on the support set S={(i,i),(i,i+1)} of size 2m, they show that any SOS decomposition must contain at least 2m orthonormal vectors, i.e., R≥2m. Consequently sos(P_{m,m,2m})=2m (Theorem 2.7). This yields a linear lower bound for the worst‑case SOS rank of square simple forms.

Lower bound for arbitrary dimensions. Lemma 2.8 and Corollary 2.9 prove a monotonicity property: adding a new variable (or a new column/row) to a simple form forces the SOS rank to increase by at least one. By induction on m+n, Theorem 2.10 shows that for all m,n≥3, the maximal SOS rank over simple m×n biquadratics satisfies BSR_simple(m,n)≥m+n, and therefore the unrestricted bound BSR(m,n)≥m+n. The construction Q_{m,n} in Example 2.11 attains exactly m+n squares, confirming the tightness of the bound when m=n.

Diagonal 3×3 forms. For diagonal PSD forms P(x,y)=∑{i,j=1}^3 a{ij}x_i^2 y_j^2 with a_{ij}≥0, the known general bound is mn−1=8. The authors improve this to 7 (Theorem 3.1). They distinguish three cases based on the number t of positive coefficients. If t≤7, each term is a single square. For t=8, one entry is zero; a 2×2 positive block W_{1212} can be expressed with at most three squares, and the remaining four terms are single squares, giving ≤7. The hardest case t=9 (all entries positive) is handled by a constructive “splitting” technique: they decompose the 2×2 block (1,2)×(1,2) into a sum of two squares (when a certain determinant c=0) or three squares otherwise, then treat the remaining 2×2 block (2,3)×(2,3) similarly, and finally add the two leftover terms. In every sub‑case the total number of squares never exceeds 7. This result sharpens the universal bound for 3×3 biquadratics.

Implications and open problems. The paper conjectures that BSR(3,3)=6, i.e., the six‑term simple form is extremal even among all 3×3 biquadratics. It also highlights the gap between the linear lower bound 2m and the general upper bound m^2−1 for square matrices, leaving open whether the true growth of BSR(m,m) is linear or quadratic. For diagonal forms, it remains unknown whether the bound 7 is attainable; no explicit coefficient matrix achieving exactly seven squares is provided.

Overall, the work demonstrates that structural restrictions (simplicity, diagonalness) can dramatically reduce the SOS rank, provides exact values for several families, and establishes robust lower bounds for the worst‑case SOS rank of biquadratic forms. These findings are relevant to real algebraic geometry, polynomial optimization, and the design of SOS‑based algorithms, where the size of the SOS certificate directly impacts computational tractability.