Counting $k$-tuples of positive integers such that the values of several polynomials of $k$ variables are relatively prime

Let $F=(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k))$ be a system of nonconstant polynomials of $k$ variables with integer coefficients and let \[ {\gcd}_F(x_1,\ldots,x_k)= \gcd(f_1(x_1,\ldots,x_k),\ldots, f_m(x_1,\ldots,x_k)). \] We obtain an unconditional asymptotic formula for the sum \[ \sum