Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$

Abstract: We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap. For simple biquadratic forms those containing only distinct terms of the type $x_i^2 y_j^2$ we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a $C_4$-free bipartite graph with the maximum number of edges. This settles the question for simple forms. For $y$-deficient biquadratic forms a class introduced here that permits cross terms among two of the three $y$-variables while the third appears only in pure square terms we prove an upper bound of $9$ by combining Calderön's theorem on $m\times 2$ forms with the known value $\mathrm{BSR}(4,2) = 5$. Our main result is a constructive proof that $\mathrm{BSR}(4,3) \geq 8$. We present an explicit non-simple, non-deficient $4\times 3$ biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than $8$ must possess a more complex algebraic structure, and it reduces the search space for determining the true value of $\mathrm{BSR}(4,3)$. Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout.