Postive Semidefinite and Sum of Squares Biquadratic Polynomials
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arXiv
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2025
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| _version_ | 1866909692354101248 |
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| author | Cui, Chunfeng Qi, Liqun Xu, Yi |
| author_facet | Cui, Chunfeng Qi, Liqun Xu, Yi |
| contents | 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 Calderón 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_17260 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Postive Semidefinite and Sum of Squares Biquadratic Polynomials Cui, Chunfeng Qi, Liqun Xu, Yi Optimization and Control 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 Calderón 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. |
| title | Postive Semidefinite and Sum of Squares Biquadratic Polynomials |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2506.17260 |