The SOS Rank of Biquadratic Forms

In 1973, Calderón proved that an $m\times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of $\frac{3m(m+1)}{2}$ squares of quadratic forms. Very recently, by applying Hilbert’s theorem on ternary quartics, we proved that a $2\times 2$ 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 $m\times 2$ biquadratic forms for $m\ge 3$ to further exploration. In this paper, we show that an $3\times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m\times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.