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.