二阶Barvinok矩阵空间的拓扑
The topology of the space of matrices of Barvinok rank two
摘要：
$d \times n$矩阵的Barvinok秩是$\mathbb{R}^d$中的点的最小数量，使得点的热带凸包包含矩阵的所有列。这个概念起源于Barvinok和其他人对旅行推销员问题的研究。我们的研究对象是实$d \times n$矩阵空间的Barvinok秩二。令$B_{d,n}$表示该空间模重新缩放和平移。我们证明了$B_{d,n}$是一个流形，从而解决了一个猜想，由于Escherin。事实上，$B_{d,n}$同胚于球面乘积的商$S^{d-2} \times S^{n-2}$在对合下，该对合使两个分量中的每个点同时到达其对极。另外，利用离散Morse理论，我们计算了$B_{d,n}$的积分同调.假设$d \ge n$，对于奇数$d$，同源性与$S^{d-2} \times \mathbb{RP}^{n-2}$同构。这对于甚至高达$d-3$度的$d$也是如此，但是这两种情况不同于$d-2$度和更高度。同调计算直接扩展到$(S^{d-2} \times X)/\mathbb{Z}_2$形式的更一般的复形，其中$X$是维数至多为$d-2$的有限胞腔复形，允许自由$\mathbb{Z}_2$作用。
英文摘要：
The Barvinok rank of a $d \times n$ matrix is the minimum number of points in $\mathbb{R}^d$ such that the tropical convex hull of the points contains all columns of the matrix. The concept originated in work by Barvinok and others on the travelling salesman problem. Our object of study is the space of real $d \times n$ matrices of Barvinok rank two. Let $B_{d,n}$ denote this space modulo rescaling and translation. We show that $B_{d,n}$ is a manifold, thereby settling a conjecture due to Develin. In fact, $B_{d,n}$ is homeomorphic to the quotient of the product of spheres $S^{d-2} \times S^{n-2}$ under the involution which sends each point to its antipode simultaneously in both components. In addition, using discrete Morse theory, we compute the integral homology of $B_{d,n}$. Assuming $d \ge n$, for odd $d$ the homology turns out to be isomorphic to that of $S^{d-2} \times \mathbb{RP}^{n-2}$. This is true also for even $d$ up to degree $d-3$, but the two cases differ from degree $d-2$ and up. The homology computation straightforwardly extends to more general complexes of the form $(S^{d-2} \times X)/\mathbb{Z}_2$, where $X$ is a finite cell complex of dimension at most $d-2$ admitting a free $\mathbb{Z}_2$-action.
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