Positive Semi-Definite 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, Ch...
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2025-07-01
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| author | Chunfeng Cui Liqun Qi Yi Xu |
| author_facet | Chunfeng Cui Liqun Qi Yi Xu |
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| description | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> biquadratic polynomial is a homogeneous quartic polynomial of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></semantics></math></inline-formula> variables. In this paper, we show that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> variables. Therefore, by Hilbert’s theorem, a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> biquadratic polynomial can be expressed as the sum of squares of nine quadratic polynomials. Furthermore, we present a necessary and sufficient condition for an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mi>n</mi></mrow></semantics></math></inline-formula>. Then we give a constructive proof of the sos form of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> psd biquadratic polynomial in three cases. |
| format | Article |
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| language | English |
| publishDate | 2025-07-01 |
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| spelling | doaj-art-b4a8154ff7cf4aeea6ea216b0c04afac2025-08-20T03:08:06ZengMDPI AGMathematics2227-73902025-07-011314229410.3390/math13142294Positive Semi-Definite and Sum of Squares Biquadratic PolynomialsChunfeng Cui0Liqun Qi1Yi Xu2School of Mathematical Sciences, Beihang University, Beijing 100191, ChinaJiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, ChinaJiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjing 211189, ChinaHilbert 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> biquadratic polynomial is a homogeneous quartic polynomial of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi></mrow></semantics></math></inline-formula> variables. In this paper, we show that an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula> variables. Therefore, by Hilbert’s theorem, a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> biquadratic polynomial can be expressed as the sum of squares of nine quadratic polynomials. Furthermore, we present a necessary and sufficient condition for an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></semantics></math></inline-formula> psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mi>n</mi></mrow></semantics></math></inline-formula>. Then we give a constructive proof of the sos form of a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo>×</mo><mn>2</mn></mrow></semantics></math></inline-formula> psd biquadratic polynomial in three cases.https://www.mdpi.com/2227-7390/13/14/2294biquadratic polynomialssum of squarespositive semi-definitenessbiquadratic polynomialstripartite quartic polynomials |
| spellingShingle | Chunfeng Cui Liqun Qi Yi Xu Positive Semi-Definite and Sum of Squares Biquadratic Polynomials Mathematics biquadratic polynomials sum of squares positive semi-definiteness biquadratic polynomials tripartite quartic polynomials |
| title | Positive Semi-Definite and Sum of Squares Biquadratic Polynomials |
| title_full | Positive Semi-Definite and Sum of Squares Biquadratic Polynomials |
| title_fullStr | Positive Semi-Definite and Sum of Squares Biquadratic Polynomials |
| title_full_unstemmed | Positive Semi-Definite and Sum of Squares Biquadratic Polynomials |
| title_short | Positive Semi-Definite and Sum of Squares Biquadratic Polynomials |
| title_sort | positive semi definite and sum of squares biquadratic polynomials |
| topic | biquadratic polynomials sum of squares positive semi-definiteness biquadratic polynomials tripartite quartic polynomials |
| url | https://www.mdpi.com/2227-7390/13/14/2294 |
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