保持非负矩阵不变的固定次数多项式锥的性质
Properties of the cone of polynomials of fixed degree that preserve nonnegative matrices
AI总结：
本文研究保持非负矩阵不变的多项式锥的几何性质，证明高次多项式生成的锥非多面体，回答了Loewy关于中心项负值程度的问题，并给出该锥在单位球面上体积的初步界。
摘要：
正如Loewy和London在[Linear and Multilinear Algebra 6 (1978/79), no. 1, 83--90]中详细阐述的那样，保持矩阵非负性的多项式锥可能在非负逆特征值问题的求解中发挥重要作用。本文首先证明，由次数大于或等于$2n$且保持$n$阶非负矩阵的多项式所生成的锥是非多面体的。其次，回答了Loewy在[Linear Algebra and its Applications, 676(2023), 267--276]中提出的一个问题，即$2n$次多项式的中心项可以负到什么程度。我们将此结果推广，证明一个保持$n$阶非负矩阵的多项式，其绝对值最大的项可以任意负，而其余系数均为1。最后，我们通过探讨该锥限制在单位球面上时的测度性质，并证明该体积的一些初始界来结束本文。
英文摘要：
As was detailed by Loewy and London in [Linear and Multilinear Algebra 6 (1978/79), no.~1, 83--90], the cone of polynomials that preserve the nonnegativity of matrices may play an important role in the solution to the nonnegative inverse eigenvalue problem. In this paper, we start by showing the cone generated by polynomials of degree greater than or equal to $2n$ that preserve nonnegative matrices of order $n$ is non-polyhedral. Next, a question posed by Loewy in [Linear Algebra and its Applications, 676(2023), 267--276], about how negative the center term can be in a degree $2n$ polynomial is answered. We extend this to show that a polynomial that preserves nonnegative matrices of order $n$ can have it's the largest term, in absolute value, be arbitrarily negative with the remaining coefficients being one. We conclude, by exploring properties of the measure of the cone when restricted to the unit sphere and by proving some initial bounds of that volume.
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