Three-Edges and the SOS Rank of Biquadratic Forms

Abstract

We extend the augmented bipartite graph framework for biquadratic sum-of-squares (SOS) ranks by introducing \emph{3-edges} -- triples of cells representing squares of three-term bilinear forms $(x_i y_j + x_k y_l + x_p y_q)^2$. The main challenge is to define suitable \emph{generalized cycle-free} conditions that are purely combinatorial yet sufficient to guarantee that the SOS rank equals the total number of edges. We give a complete definition that carefully distinguishes occupation by $1$/$2$-edges from occupation by $3$-edges, and introduce a separate condition for $3$-edges. The main theorem states that for any generalized cycle-free augmented bipartite graph $G$ satisfying the simplicity condition (S), the associated \emph{triply simple biquadratic form} $P_G$ satisfies $\operatorname{sos}(P_G) = |E_1| + |E_2| + |E_3|$. The proof extends the orthogonality method with a novel trick: when a $2$-edge and a $3$-edge interact, the $3$-edge condition must be invoked rather than the $2$-edge condition. As concrete applications, we construct a $10 \times 5$ graph using a column-fully-degenerate $3$-edge, showing $z_{3L}(10,5) \ge 27$ and $\operatorname{BSR}(10,5) \ge 27$, which separates $z_{3L}(10,5)$ from $z_L(10,5)=26$; and a $15 \times 6$ graph using a half-row-degenerate $3$-edge, improving the lower bound for $\operatorname{BSR}(15,6)$ from $43$ to $44$. These are the first explicit applications of $3$-edges (both fully degenerate and half-degenerate) to obtain improved lower bounds for $\operatorname{BSR}(m,n)$.