Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

TL;DR

This paper investigates the sum of squares rank of biquadratic forms, establishing a lower bound related to the Zarankiewicz number and conjecturing their equality, thus linking graph theory and SOS polynomial theory.

Contribution

The paper proves a lower bound for the SOS rank of biquadratic forms and conjectures its exact value equals the Zarankiewicz number, extending known results and connecting two mathematical fields.

Findings

Established that BSR(m, n) ≥ z(m, n)

Confirmed the result for specific cases m=2, n=2 and m=n=3

Proposed the conjecture BSR(m, n) = z(m, n)

Abstract

Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $BSR(m, n)$. In this paper, we show that $BSR(m, n) \ge z(m, n)$ and conjecture that $BSR(m, n) = z(m, n)$, 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.