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$\mathcal L$ 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 $\min(m,m')$?
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