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Take $n,d,B\in\mathbb Z_{>0}$ with $d<n$ and denote $\mathcal M_{n,d}$ to be set of all total degree $d$ monomials in $n$ variables $x_1,\dots,x_n$ with degree $\leq1$ in each variable (monomials only of form $x_1\cdot\dots\cdot x_d$).

Denote $X$ to be vector of all such monomials in $\mathcal M_{n,d}$. Take integer vectors $v_1,\dots,v_t$ of length $|\mathcal M_{n,d}|$ with each entry taken uniformly and independently from $[-B,B]\cap\mathbb Z\setminus\{0\}$.

Consider the $t$ polynomials $v_1X'+p_1,\dots,v_tX'+p_t\in\mathbb Z[x_1,\dots,x_n]$ at random distinct primes $p_1,\dots,p_t\in[-B,B]$ where $'$ denotes transpose.

What is the probability that at every $1\leq i<j\leq t$ we have $$\mathsf{deg}(\mathsf{gcd}(v_iX'+p_i,v_jX'+p_j))=0?$$

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