The SOS Rank of Biquadratic Forms

In 1973, Calderón proved that an m × 2 positive semidefinite (psd) biquadratic form can always be expressed as the sum of 3m(m+1) 2 squares of quadratic forms. Very recently, by applying Hilbert's theorem, we proved that a 2 × 2 psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calderón's result for $m=2$, and left the sos (sum-of-squares) rank problem of m × 2 biquadratic forms for m ≥ 3 to further exploration. In this paper, we show that a 3 × 2 psd biquadratic form can always be expressed as four squares of bilinear forms. For m ≥ 4, we make a conjecture that an m × 2 psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.