Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
Linear Principal Minor Polynomials: Hyperbolic Determinantal Inequalities and Spectral Containment
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DOI：
10.1093/imrn/rnac291
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发表时间：
2021-12
影响因子：
1
通讯作者：
Grigoriy Blekherman;Mario Kummer;Raman Sanyal;Kevin Shu;Shengding Sun
中科院分区：
文献类型：
--
作者：
Grigoriy Blekherman;Mario Kummer;Raman Sanyal;Kevin Shu;Shengding Sun
线性主次多项式或lpm多项式是对称矩阵的主次多项式的线性组合。通过限制在对角线上，lpm多项式与多仿射多项式是双射的。我们发现，这建立了一个一对一的对应关系之间的齐次多仿射稳定多项式和PSD稳定的lpm多项式。这产生了新的双曲多项式的建设技术，并允许我们找到一个明确的3度双曲多项式在6个变量的瑞利差异不是平方和。我们进一步将著名的Fisher-Hadamard和Koteljanskii不等式从行列式推广到PSD稳定的lpm多项式。我们调查相关的双曲锥和猜想之间的关系的特征值的对称矩阵和某些lpm多项式的值在该矩阵的关系。我们把这种关系称为光谱包容。
A linear principal minor polynomial or lpm polynomial is a linear combination of principal minors of a symmetric matrix. By restricting to the diagonal, lpm polynomials are in bijection with multiaffine polynomials. We show that this establishes a one-to-one correspondence between homogeneous multiaffine stable polynomials and PSD-stable lpm polynomials. This yields new construction techniques for hyperbolic polynomials and allows us to find an explicit degree 3 hyperbolic polynomial in six variables some of whose Rayleigh differences are not sums of squares. We further generalize the well-known Fisher–Hadamard and Koteljanskii inequalities from determinants to PSD-stable lpm polynomials. We investigate the relationship between the associated hyperbolicity cones and conjecture a relationship between the eigenvalues of a symmetric matrix and the values of certain lpm polynomials evaluated at that matrix. We refer to this relationship as spectral containment.