正双曲簇、热带化与正拟阵
Positively Hyperbolic Varieties, Tropicalization, and Positroids
摘要：
复仿射空间中的余维簇$c$称为正双曲的，如果其中任意点的虚部不位于维数为$c$的任何正线性子空间中。正双曲超曲面由稳定多项式定义。我们给出了一个新的特征的正双曲品种使用符号变化，并表明，他们是等价的定义是双曲相对于正部分的Grassmannian，在意义上的Shamovich和Vinnikov。证明了正双曲射影簇的热带化是由$x_i = x_j$定义的$A$型超平面安排的局部子扇，其中最大锥满足非交叉条件。这给出了新的证明的一些结果Choe-奥克斯利-索卡尔-瓦格纳和Brändén牛顿多面体和热带化的稳定多项式。我们解决的问题，哪些热带品种可以得到的热带化的正双曲品种的情况下，热带环面品种，常数系数热带曲线，和伯格曼球迷。在此过程中，我们还给出了一个新的特征的positroids在一个非交叉条件的伯格曼球迷。
英文摘要：
A variety of codimension $c$ in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension $c$. Positively hyperbolic hypersurfaces are defined by stable polynomials. We give a new characterization of positively hyperbolic varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. We prove that positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type $A$ hyperplane arrangement defined by $x_i = x_j$, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of some results of Choe--Oxley--Sokal--Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we also give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.
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