On the Bhargava factorial of polynomial maps

Bhargava introduced a generalization of the factorial function to extend classical results in integers to Dedekind rings and unify them. We study the Bhargava factorial of the images of polynomial maps from an analytic perspective. We first give the $\mathfrak{p}$-adic closures of the images of polynomial maps, which is the key to compute $\mathfrak{p}$-adic part of the Bhargava factorial. Then, as a special case, we give the Stirling formula for the image of quadratic polynomials with integer coefficients.