Positive Semi-Definite and Sum of Squares Biquadratic Polynomials

Chunfeng Cui, Liqun Qi () and Yi Xu
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Chunfeng Cui: School of Mathematical Sciences, Beihang University, Beijing 100191, China
Liqun Qi: Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjin 211189, China
Yi Xu: Jiangsu Provincial Scientific Research Center of Applied Mathematics, Nanjin 211189, China

Mathematics, 2025, vol. 13, issue 14, 1-15

Abstract: 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 × n biquadratic polynomial is a homogeneous quartic polynomial of m + n variables. In this paper, we show that an m × n biquadratic polynomial can be expressed as a tripartite homogeneous quartic polynomial of m + n − 1 variables. Therefore, by Hilbert’s theorem, a 2 × 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 × 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 × n psd biquadratic polynomial to be sos, and show that if such a polynomial is sos, then its sos rank is at most m n . Then we give a constructive proof of the sos form of a 2 × 2 psd biquadratic polynomial in three cases.

Keywords: biquadratic polynomials; sum of squares; positive semi-definiteness; biquadratic polynomials; tripartite quartic polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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