Biquadratic SOS Rank and Augmented Zarankiewicz Number

This paper introduces the concepts of the augmented Zarankiewicz number zA(m,n) and the limited augmented Zarankiewicz number zL(m,n), which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers arise from augmented bipartite graphs that may contain both standard edges (1-edges) and pairs of edges representing squares of binomials (2-edges). The main theoretical result establishes the inequality chain BSR(m,n)≥zA(m,n)≥zL(m,n)≥z(m,n), linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of zL(m,n) for the cases (m,2), (3,3), (4,3), and (4,4) and provide new lower bounds for the cases (5,3), (5,4), and (5,5). These results yield improved lower bounds for the maximum SOS rank of biquadratic forms, demonstrating that zL(m,n) can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem.