Postive Semidefinite and Sum of Squares Biquadratic Polynomials

Hilbert proved in 1888 that a positive semi-definite (PSD) homogeneous quartic polynomial of three variables always can be expressed as the sum of squares (SOS) of three quadratic polynomials, and a psd homogeneous quartic polynomial of four variables may not be sos. Only after 87 years, in 1975, Choi gave the explicit expression of such a psd-not-sos (PNS) homogeneous quartic polynomial of four variables. An $m \times n$ biquadratic polynomial is a homogeneous quartic polynomial of $m+n$ variables. In this paper, we show that an $m \times n$ biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of $m+n-1$ variables. Therefore, {by Hilbert's theorem}, a $2 \times 2$ PSD biquadratic polynomial can be expressed as the sum of squares of three quadratic polynomials. This improves the result of Calder\'{o}n in 1973, who proved that a $2 \times 2$ biquadratic polynomial can be expressed as the sum of squares of nine quadratic polynomials. Furthermore, we present a necessary and sufficient condition for an $m \times n$ psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most $mn$. Then we give a constructive proof of the sos form of a $2 \times 2$ psd biquadratic polynomial in three cases.