On the SOS Rank of Simple and Diagonal Biquadratic Forms

TL;DR

This paper investigates the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms, establishing exact maximum ranks in certain cases and bounds that reveal how structure influences the minimal number of squares needed.

Contribution

It provides new exact values and bounds for the SOS rank of biquadratic forms, demonstrating how form structure affects the number of squares required.

Findings

Maximum SOS rank for 3x3 simple biquadratic forms is 6.

Existence of m x m simple biquadratic forms with SOS rank exactly 2m.

Upper bound of 7 for SOS rank of diagonal biquadratic forms with nonnegative coefficients.

Abstract

We study the sum-of-squares (SOS) rank of simple and diagonal biquadratic forms. For simple biquadratic forms in $3 \times 3$ variables, we show that the maximum SOS rank is exactly $6$, attained by a specific six-term form. We further prove that for any $m \ge 3$, there exists an $m \times m$ simple biquadratic form whose SOS rank is exactly $2m$. Moreover, we show that for all $m, n \ge 3$, the maximum SOS rank over $m \times n$ simple biquadratic forms is at least $m+n$, which implies $\mathrm{BSR}(m,n) \ge m+n$. For diagonal biquadratic forms with nonnegative coefficients, we prove an SOS rank upper bound of $7$, improving the general bound of $8$ for $3 \times 3$ forms. These results provide new lower and upper bounds on the worst-case SOS rank of biquadratic forms and highlight the role of structure in reducing the required number of squares.