The SOS Rank of a $5 \times 4$ Biquadratic Form via Orthogonality

Abstract

Biquadratic forms arise naturally in polynomial optimization, tensor analysis, and quantum information theory. A key problem is determining the minimal number of squares needed in a sum-of-squares (SOS) representation of such a form, known as its SOS rank. For fixed dimensions $(m,n)$, the maximum possible SOS rank over all biquadratic forms in $m$ and $n$ variables is denoted $\operatorname{BSR}(m,n)$. Recent advances have established lower bounds on $\operatorname{BSR}(m,n)$ via combinatorial constructions involving bipartite graphs and the orthogonality method. In particular, for the case $(m,n)=(5,4)$, it was shown that $\operatorname{BSR}(5,4)\ge 11$ using only nondegenerate 2-edges. In this paper, we extend this framework by incorporating a degenerate $2$-edge, which introduces a cross term where the two $y$-indices coincide. We construct an explicit $5\times 4$ biquadratic form and apply the orthogonality method to prove that its SOS rank is $12$, thereby improving the lower bound to $\operatorname{BSR}(5,4)\ge12$. This result demonstrates that degenerate $2$-edges yield additional algebraic flexibility beyond purely combinatorial bounds and extends the applicability of the orthogonality method to forms with cross terms involving identical $y$-indices.