Biquadratic Tensors and Biquadratic Polynomials

祁力群，中国著名数学家，曾任教于清华大学、澳大利亚新南威尔士大学、香港城市大学等高校，还曾为香港理工大学讲席教授和应用数学系系主任，现为香港理工大学应用数学系荣休教授、杭州电子科技大学特聘教授。

他的主要研究工作涵盖优化及其应用、计算数学及其应用两个方面。在优化领域，他在广义 Newton 方法、非线性规划与变分不等式方面有开创性研究；在计算数学领域，开创了张量计算研究的多个方面工作，并在数值多线性代数与非线性方程组方面成绩显著。2005 年他首次引入高阶张量的特征值概念，这些理论在医学工程、统计数据分析以及固体力学等领域得到广泛应用。

祁力群教授获得了诸多荣誉，如 1981 年至 1999 年成为 ISI 高引用科学家，被誉为 1981-1999 年间国际最具影响力数学家之一；2003 年 - 2004 年因科学研究与学术活动的杰出工作获得香港理工大学校长奖；2010 年荣获第二届中国运筹学会科学技术奖一等奖等。

他在学术成果方面成果丰硕，自 1985 年以来，主持了数十个科学研究项目，组织了几十次国际会议，在三十多个国际会议上做大会报告或邀请报告，出版或主编论著与专刊近二十部，发表高级别论文二百余篇。此外，他还是 8 个国际期刊的主编或者编委，并曾任国内外二十多所大学或研究机构的访问教授或客座研究员。

In this talk, I will report some of our recent work on biquadratic tensors and biquadratic polynomials, and some challenging open problems related with our research.

The first is the nonnegative biquadratic tensor problem. An M-eigenvalue of a nonnegative biquadratic tensor is referred to as an M+-eigenvalue if it has a pair of nonnegative M-eigenvectors. If furthermore that pair of M-eigenvectors is positive, then thatM+-eigenvalue is called an M++-eigenvalue. A nonnegative biquadratic tensor has atleast one M+ eigenvalue, and the largest M+-eigenvalue is both the largest M-eigenvalueand the M-spectral radius. For irreducible nonnegative biquadratic tensors, all the M+-eigenvalues are M++-eigenvalues. For an irreducible nonnegative biquadratic tensor, thelargest M+-eigenvalue has a max-min characterization, while the smallest M+-eigenvaluehas a min-max characterization.

The first open problem is: How can we construct an algorithm to compute the Mspectral radius of a nonnegative biquadratic tensor, and establish its convergence.

The second is the psd and sos problem of biquadratic polynomials. Hilbert proved in1888 that a positive semi-definite (psd) homogeneous quartic polynomial of three variablesalways can be expressed as the sum of squares (sos) of three quadratic polynomials, anda psd homogeneous quartic polynomial of four variables may not be sos. An m × nbiquadratic polynomial is a homogeneous quartic polynomial of m + n variables. Weshow that an m ×n biquadratic polynomial can be expressed as a tripartite homogeneousquartic polynomial of m + n − 1 variables. Therefore, by Hilbert’s theorem, a 2 × 2 psdbiquadratic polynomial can always be expressed as the sum of squares of three quadraticpolynomials. This improves the result of Calder´on in 1973, who proved that an m × 2PSD biquadratic polynomial can be expressed as the sum of squares of 3m(m+1)/2 quadratic polynomials.

Hence, an open problem is what is the SOS rank for an m × 2 PSD biquaratic polynomial for m ≥ 3. In 1975, Choi gave a 3 × 3 PNS biquadratic polynomial. Another open problem is: What is the maximum SOS rank of an m × n SOS biquadratic polynomialfor m, n ≥ 3. An even more challenging open problem is to determine a given m × n PSDbiquadratic polynomial is SOS or not numerically. Recently, we proved that a psd 3 × 2biquadratic form can be expressed by four squares of bilinear forms.