Take vector $u=\begin{bmatrix}x_1,\dots,x_n\end{bmatrix}\otimes\begin{bmatrix}y_1,\dots,y_n\end{bmatrix}$ where $x_i,y_j$ are variables and consider the vector $v=\underbrace{u\otimes\dots\otimes u}_{d\mbox{ times}}$. It contains monomials in variables $x_i,y_j$ and not all monomials occur.
Take $r$ polynomials in $\mathbb Z[x_1,\dots,x_n,y_1,\dots,y_n]$ with monomials only from entries in vector $v$.
Is there a simple condition of algebraic independence of $r$ such polynomials?