Let $A\in\{0,1\}^{n\times n}$ of real rank $r$.
Let $J$ be all one matrix.
Denote $\underline{x}=(x_1,\dots,x_n)$, $\underline{y}=(y_1,\dots,y_n)$.
It is clear we have $(\sum_{i}x_i)(\sum_jy_j)=\underline{x}J\underline{y}=\underline{x}(A+J-A)\underline{y}=(\underline{x}A + \underline{x}(J-A))\underline{y}=\underline{x}A\underline{y} + \underline{x}(J-A)\underline{y}$.
Let $L_1,L_2$ be minimum possible such that: $$\underline{x}A\underline{y}=\sum_{t=1}^{L_1}f_{1,t}(\underline{x})g_{1,t}(\underline{y})$$ $$\underline{x}(J-A)\underline{y}=\sum_{t=1}^{L_2}f_{2,t}(\underline{x})g_{2,t}(\underline{y})$$
where coefficients of $f_{i,t}(\underline{x}),g_{i,t}(\underline{x})\in\Bbb Z[\underline{x},\underline{y}]$ are of just $\{0,1\}$ value.
Let $L_{max}=\max(L_1,L_2)$, $L_{min}=\min(L_1,L_2)$.
Show that $$\log r\leq\log L_{min}\leq \log L_{max}\leq (\log L_{min})^c\leq(\log r)^d$$ where $c,d>1$ is fixed holds with every $A\in\{0,1\}^{n\times n}$ of real rank $r$.