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

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.