  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.
