Seminars on Numerical Algebra, Optimization and Data Sciences：The SOS Problem of Biquadratic Forms

Speaker：Liqun Qi (The Hong Kong Polytechnic University)

Time：2025-12-31 16:00

Location：Conference Room C503 at Administration Building at Haiyun Campus

Abstract:

A fundamental question at the intersection of algebra and optimization is whether a multivariate polynomial that is nonnegative everywhere (positive semi-definite, or PSD) can be written as a sum of squares (SOS) of polynomials. This talk focuses on the SOS Problem of a specific and important class of polynomials: Biquadratic Forms.

In 1973, Calderon proved that an m x 2 psd biquadratic form can always be expressed as the sum of squares (sos) of 3m(m+1)/2 quadratic forms. In 1975, Choi gave a concrete example of a 3 x 3 psd biquadratic form which is not sos. This gave a general picture of the SOS problem of biquadratic forms. Recently, in a series of papers, we systematically renew our knowledge on this problem. What we have done are as follow:

1. By applying Hilbert’s theorem, we proved that a 2 x 2 psd biquadratic form can always be expressed as the sum of three squares of bilinear forms,

2. By combining real analysis and algebraic geometry, we proved that a 3 x 2 psd biquadratic form can always be expressed as the sum of four squares of bilinear forms. We further made a conjecture that a m x 2 psd biquadratic form can always be expressed as the sum of m + 1 squares of bilinear forms, These systematically improved Calderon’s result.

3. We introduced symmetric biquadratic forms. Choi’s example is not a symmetric biquadratic form. We showed that all psd symmetric biquadratic forms are sos. This opens a new research direction.

4. We identified a number of psd structured biquadratic tensors are sos.

This also open a new research direction.