Horn问题、Vinnikov曲线和蜂巢锥
Horn's Problem, Vinnikov Curves and the Hive Cone
摘要：
Vinnikov曲线是一条射影平面曲线，对于X，Y和Z正定Hermitian n × n矩阵，可以写成det（xX+yY+zZ）=0的形式。给定正实数的三个n元组alpha，beta和gamma，GL_n \CC中存在A，B和C，且具有奇异值alpha，beta和gamma，ABC=1当且仅当存在一条Vinnikov曲线通过3 n个点（-1：alpha_i^2：0），（0：-1：beta_i^2）和（gamma_i^2：0：-1）. Knutson和Tao证明了这样的A、B和C存在的另一个等价条件是存在一个边界为（log alpha，log beta，log gamma）的蜂巢（定义于）。F的系数近似地形成了这样一个蜂巢，这就给出了Knutson和Tao的结果的一个新的证明。本文没有使用表示论，也基本上没有使用辛几何。取而代之的是，它使用了Viro的拼接方法和Vinnikov曲线的拓扑描述。
英文摘要：
A Vinnikov curve is a projective plane curve which can be written in the form det(xX+yY+zZ)=0 for X, Y and Z positive definite Hermitian n by n matrices. Given three n-tuples of positive real numbers, alpha, beta and gamma, there exist A, B and C in GL_n \CC with singular values alpha, beta and gamma and ABC=1 if and only if there is a Vinnikov curve passing through the 3n points (-1: alpha_i^2:0), (0:-1:beta_i^2) and (gamma_i^2:0:-1). Knutson and Tao proved that another equivalent condition for such A, B and C to exist is that there is a hive (defined within) whose boundary is (log alpha, log beta, log gamma). The logarithms of the coefficients of F approximately form such a hive; this leads to a new proof of Knutson and Tao's result. This paper uses no representation theory and essentially no symplectic geometry. In their place, it uses Viro's patchworking method and a topological description of Vinnikov curves.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
