# Probability of degree $0$ gcd between every pair of random homogeneous polynomials shifted by random primes?

Take $$n,d,B\in\mathbb Z_{>0}$$ with $$d and denote $$\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 $$1\leq i we have $$\mathsf{deg}(\mathsf{gcd}(v_iX'+p_i,v_jX'+p_j))=0?$$