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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2025-07-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/14/2294 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |