Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

We study positive semi-definite (PSD) biquadratic forms and their sum-of-squares (SOS) representations. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is an SOS. This extends the known result for fully symmetric biquadratic forms. Furthermore, we describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For m≥2, we provide a explicit example to show the lower bound for sos rank of m×2 biquadratic forms is m+1, and show that previously proved results indicating that a 2×2 PSD biquadratic form can be expressed as the sum of three squares and a 3×2 PSD biquadratic form can be expressed as the sum of four squares are tight. We also present an 3×3 SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares. Moreover, we establish a universal upper bound mn−1 for any m×n SOS biquadratic form, which improves the trivial bound mn.