Biquadratic SOS Rank and Double Zarankiewicz Number

Abstract: The maximum sum-of-squares (SOS) rank of an $m\times n$ biquadratic form, denoted $\operatorname{BSR}(m,n)$, has recently been shown to be bounded below by the classical Zarankiewicz number $z(m,n)$, the maximum number of edges in a $C_4$-free bipartite graph. However, a gap was discovered for the $4\times3$ case: $\operatorname{BSR}(4,3)\ge 8 > z(4,3)=7$, arising from a construction that adds a square of a bilinear form $(x_4y_2+x_1y_3)^2$ to a $C_4$-free simple biquadratic form. This paper introduces the double Zarankiewicz number $z_2(m,n)$ to capture such phenomena. We define a new class of bipartite graphs that may contain both ordinary edges (1-edges) and ``double edges'' (2-edges), the latter corresponding to squares of simple two-term bilinear forms. A graph $G$ of this type yields a doubly simple biquadratic form $P_G$, and we prove that if $G$ contains no \emph{generalised $C_4$-cycle} then $\operatorname{sos}(P_G)=|E_1|+|E_2|$; consequently $\operatorname{BSR}(m,n)\ge z_2(m,n)$. We determine the exact value of $z_2(m,n)$ for several small parameters: $z_2(m,2)=m+1$, $z_2(2,n)=n+1$, $z_2(3,3)=6$, $z_2(4,3)=8$, and $z_2(5,3)=9$. For the $4\times4$ case we establish $10\le z_2(4,4)\le 11$, leaving the exact value as an open problem. These results improve the known lower bounds for $\operatorname{BSR}(4,4)$ and $\operatorname{BSR}(5,3)$. The paper concludes with a discussion of open questions and connections between extremal graph theory and the study of SOS representations.