We show that any bounded integral function $f : A \times B \mapsto \{0,1, \dots, \Delta\}$ with rank $r$ has deterministic communication complexity $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log^2 r$, where the rank of $f$ is defined to be the rank of the $A \times B$ matrix whose entries are the function values. As a corollary, we show that any $n$-dimensional polytope that admits a slack matrix with entries from $\{0,1,\dots,\Delta\}$ has extension complexity at most $\exp(\Delta^{O(\Delta)} \cdot \sqrt{n} \cdot \log^2 n)$.

翻译：我们证明，任意有界整数函数 $f : A \times B \mapsto \{0,1, \dots, \Delta\}$，其秩为 $r$，则具有确定性通信复杂度 $\Delta^{O(\Delta)} \cdot \sqrt{r} \cdot \log^2 r$，其中 $f$ 的秩定义为以函数值为元素的 $A \times B$ 矩阵的秩。作为推论，我们证明任意 $n$ 维多面体，若其松弛矩阵的条目取自 $\{0,1,\dots,\Delta\}$，则其扩展复杂度至多为 $\exp(\Delta^{O(\Delta)} \cdot \sqrt{n} \cdot \log^2 n)$。