Biquadratic SOS Rank and Augmented Zarankiewicz Number

TL;DR

This paper introduces augmented Zarankiewicz numbers, linking them to SOS rank, and determines exact values for specific cases, providing new bounds and insights into biquadratic forms.

Contribution

It defines new combinatorial parameters extending Zarankiewicz numbers and establishes their relation to SOS rank, with exact calculations and bounds for key cases.

Findings

Exact values of $z_L(m,n)$ for (m,2), (3,3), (4,3), (4,4)

New lower bounds for $z_L(m,n)$ in cases (5,3), (5,4), (5,5)

Improved lower bounds for maximum SOS rank of biquadratic forms

Abstract

This paper introduces the concepts of the augmented Zarankiewicz number $z_A(m,n)$ and the limited augmented Zarankiewicz number $z_L(m,n)$, 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 $\operatorname{BSR}(m, n) \geq z_A(m, n) \geq z_L(m, n) \geq z(m, n)$, linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of $z_L(m, n)$ 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 $z_L(m,n)$ can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem.