Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

Chunfeng Cui; Liqun Qi; Yi Xu

paper

Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

arXiv:2602.07844

Abstract

Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $B S R (m, n)$ . In this paper, we show that $B S R (m, n) \geq z (m, n)$ and conjecture that $B S R (m, n) = z (m, n)$ , where $z (m, n)$ is the Zarankiewicz number. Our result coincides with the existing results for $m = 2$ , $n = 2$ , and $m = n = 3$ , and is superior to other previously known lower bounds. Our result also connects graph theory and SOS polynomial theory.