Abstract
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 -adic closures of the images of polynomial maps, which is the key to compute
-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.
Acknowledgments
The author would like to express his sincere gratitude to the referees for several helpful suggestions which led to the improvement of this paper.