论文标题
多维矩阵的多维永久物
Multidimensional permanents of polystochastic matrices
论文作者
论文摘要
a $ d $维矩阵称为\ emph {$ 1 $ -polystochastic}，如果它是非负的，并且每行的总和等于〜$ 1 $。这样的矩阵在每行中有单个$ 1 $，而其他地方的零为零，被称为\ emph {$ 1 $ - permuntion}矩阵。 $ d $二维矩阵$ n $的\ emph {对角线}是$ n $元素的选择，在同一超平面中没有两个。 $ d $维矩阵的\ emph {permanent}是对角线内元素乘积的对角线的总和。 对于给定的订单$ n $和dimension $ d $，$ 1 $ - PolyStochantic矩阵的集合形成了一个凸polytope，其中包括$ 1 $ - permuntion矩阵的矩阵。对于甚至$ n $和奇数$ d $，我们为零永久的$ 1 $ permunt矩阵提供了建设。因此，我们表明，永久零的$ 1 $ -Polystochastic矩阵至少包含$ n^{n^{n^{3/2}（1/2-o（1）} $ $ 1 $ 1 $ - perm-perm-perm-perm-perm-perm-perm-perm-perm-perm-perm-perm-perm-cn^{3/2} $ for $ cn^{3/2} $ for $ c，d $ c，d $ c，以及我们还为Taranenko的猜想提供了反示例。 对于奇数$ d $，我们提供了$ 1 $ permuart的矩阵的结构，将分解成凸线的阳性对角线总和。这些结合了Taranenko的定理，为Dow和Gibson Personizating Van der Waerden的猜想提供了反示例。
A $d$-dimensional matrix is called \emph{$1$-polystochastic} if it is non-negative and the sum over each line equals~$1$. Such a matrix that has a single $1$ in each line and zeros elsewhere is called a \emph{$1$-permutation} matrix. A \emph{diagonal} of a $d$-dimensional matrix of order $n$ is a choice of $n$ elements, no two in the same hyperplane. The \emph{permanent} of a $d$-dimensional matrix is the sum over the diagonals of the product of the elements within the diagonal. For a given order $n$ and dimension $d$, the set of $1$-polystochastic matrices forms a convex polytope that includes the $1$-permutation matrices within its set of vertices. For even $n$ and odd $d$, we give a construction for a class of $1$-permutation matrices with zero permanent. Consequently, we show that the set of $1$-polystochastic matrices with zero permanent contains at least $n^{n^{3/2}(1/2-o(1))}$ $1$-permutation matrices and contains a polytope of dimension at least $cn^{3/2}$ for fixed $c,d$ and even $n\to\infty$. We also provide counterexamples to a conjecture by Taranenko about the location of local extrema of the permanent. For odd $d$, we give a construction of $1$-permutation matrices that decompose into a convex linear sum of positive diagonals. These combine with a theorem of Taranenko to provide counterexamples to a conjecture by Dow and Gibson generalising van der Waerden's conjecture to higher dimensions.
