On the SOS Rank of Simple and Diagonal Biquadratic Forms
Yi Xu, Chufeng Cui, Liqun Qi
TL;DR
The paper investigates the SOS rank of structured biquadratic forms, focusing on simple and diagonal classes. It establishes that the maximum SOS rank for $3\times3$ simple biquadratic forms is $6$, and provides a linear lower bound $2m$ for $m\times m$ simple forms; for $3\times3$ diagonal forms with nonnegative coefficients, the SOS rank is bounded by $7$, improving the general bound. The authors develop a sparse-form framework with combinatorial arguments, including a rectangle extraction lemma, to derive upper bounds from support and construct high-rank sparse examples, showing the SOS rank can grow with the number of terms. They also connect these results to Gram-matrix sparsity, discuss open problems (notably $\mathrm{BSR}(3,3)$), and extend the methods to broader sparse classes, highlighting how structure reduces the number of squares needed in SOS representations.
Abstract
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 $2m$, providing a general lower bound that extends the $3\times3$ case. 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. In addition, we extend the techniques to a broader class of \textbf{sparse biquadratic forms}, obtaining combinatorial upper bounds and constructing explicit families whose SOS rank grows linearly with the number of terms. 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.