The SOS Rank of Biquadratic Forms

In 1973, Calder\'{o}n proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ 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 Calder\'{o}n'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.