3 problems

Let a biquadratic form be a homogeneous polynomial of degree two in each of two variable vectors, and let its SOS rank be the smallest number of squares of bilinear forms in a sum-…

Let $m,n\geq 2$. Denote by $BSR(m,n)$ the maximum sum-of-squares rank of an $m\times n$ biquadratic form, and let $z(m,n)$ be the maximum number of edges in a bipartite graph with…

Let $P(\mathbf{x},\mathbf{y})$ be a $3 \times 3$ symmetric PSD biquadratic form, so $P(\mathbf{x},\mathbf{y})\geq 0$ for all $\mathbf{x},\mathbf{y}\in\mathbb{R}^3$. A bilinear form…