summary:A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set $\{+, -, 0\}$ ($ \{ +, 0 \}$, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix $\cal A$ is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of $\cal A$. Using a correspondence between sign patterns with minimum rank $r\geq 2$ and point-hyperplane configurations in $\mathbb R^{r-1}$ and Steinitz's theorem on the rational realizability of 3-polytopes, it is shown that for every nonnegative sign pattern of minimum rank at most 4, the minimum rank and the rational minimum rank are equal. But there are nonnegative sign patterns with minimum rank 5 whose rational minimum rank is greater than 5. It is established that every $d$-polytope determines a nonnegative sign pattern with minimum rank $d+1$ that has a $(d+1)\times (d+1)$ triangular submatrix with all diagonal entries positive. It is also shown that there are at most $\min \{ 3m, 3n \}$ zero entries in any condensed nonnegative $m \times n$ sign pattern of minimum rank 3. Some bounds on the entries of some integer matrices achieving the minimum ranks of nonnegative sign patterns with minimum rank 3 or 4 are established.
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机译：摘要：符号模式矩阵（或非负符号模式矩阵）是一个矩阵，其条目来自集合$ \ {+，-，0 \} $（分别为$ \ {+，0 \} $）。符号模式矩阵$ \ cal A $的最小秩（或有理最小秩）是其符号具有等于$ \ cal A $对应条目的矩阵（分别为理性矩阵）的秩的最小值。使用最小秩$ r \ geq 2 $的符号模式与$ \ mathbb R ^ {r-1} $中的点超平面配置之间的对应关系以及Steinitz关于3多面体的合理可实现性的定理，表明最小秩为4的非负符号模式，最小秩和有理最小秩相等。但是，存在具有最小秩5的非负符号模式，其有理最小秩大于5。可以确定的是，每个$ d $ -polytope确定具有$（d + 1）的最小秩$ d + 1 $的非负符号模式。 ）\（d + 1）$三角形子矩阵，所有对角线项均为正。还表明，在最小秩3的任何压缩的非负$ m \ times n $符号模式中，最多存在$ \ min \ {3m，3n \} $个零条目。建立具有最小等级3或4的非负符号模式的最小等级。
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