# Algebraically independent vectors in tensor product

$$\mathcal L$$ and $$\mathcal L'$$ be full rank lattices in $$\mathbb R^n$$ with shortest vectors $$v_1,\dots,v_n$$ and $$v_1',\dots,v_n'$$ respectively where $$\|v_1\|_2\leq\dots\leq\|v_n\|_2$$ $$\|v_1'\|_2\leq\dots\leq\|v_n'\|_2$$ hold and $$\mathcal L=Span(v_1,\dots,v_n)$$ and $$\mathcal L'=Span(v_1',\dots,v_n')$$.

We consider the vectors of $$\mathcal L$$ and $$\mathcal L'$$ to be coefficients of a polynomial of some appropriate degree $$d$$ in $$\mathbb Z[x_1,\dots,x_k]$$ with monomials fixed in some order. Two polynomials are algebraically independent if their $$\mathsf{GCD}$$ is degree $$0$$.

Suppose $$\mathcal L$$ contains $$m$$ algebraically independent vectors with norm at most $$\|v_n\|_2$$ and $$\mathcal L'$$ contains $$m'$$ algebraically independent vectors with norm at most $$\|v'_n\|_2$$. Then $$r$$ the number of algebraically independent vectors with norm at most $$\|v_n\|_2\|v'_n\|_2$$ in $$\mathcal L\otimes\mathcal L'$$ at least $$\min(m,m')$$ .

1. Is $$m,m'$$ bound by $$n$$?
2. Is $$r$$ bound by $$\min(m,m')$$?