论文标题
决定性表示和主要次要图的图像
Determinantal representations and the image of the principal minor map
论文作者
论文摘要
在本文中，我们探讨了多芬多项式的决定性表示，以及对主要次要图下各个矩阵图像的后果。我们表明，当且仅当其所有所谓的瑞利差异因素作为Hermitian As Hermitian As Squares时，并以这种表征的结论是，Hermitian矩阵空间的形象在原始的许多方程和无限范围的轨道下，$（ sl} _2（\ mathbb {r}））^{n} \ rtimes s_ {n} $。我们还研究了具有二次扩展的更多一般领域。瑞利差异的因素化证明是捕获主要次要图的微妙行为的有效工具。与Hermitian案例相反，我们举例说明了任何字段$ \ mathbb {f} $的例子，没有有限的方程组，其在$（{\ rm sl} _2（\ rm sl} _2（\ mathbb {f}）_2（\ mathbb {f}））下$ \ Mathbb {f} $在每$ n $的主要次要地图下。
In this paper we explore determinantal representations of multiaffine polynomials and consequences for the image of various spaces of matrices under the principal minor map. We show that a real multiaffine polynomial has a definite Hermitian determinantal representation if and only if all of its so-called Rayleigh differences factor as Hermitian squares and use this characterization to conclude that the image of the space of Hermitian matrices under the principal minor map is cut out by the orbit of finitely many equations and inequalities under the action of $({\rm SL}_2(\mathbb{R}))^{n} \rtimes S_{n}$. We also study such representations over more general fields with quadratic extensions. Factorizations of Rayleigh differences prove an effective tool for capturing subtle behavior of the principal minor map. In contrast to the Hermitian case, we give examples to show for any field $\mathbb{F}$, there is no finite set of equations whose orbit under $({\rm SL}_2(\mathbb{F}))^{n} \rtimes S_{n}$ cuts out the image of $n\times n$ matrices over $\mathbb{F}$ under the principal minor map for every $n$.
