On the Bhargava factorial of polynomial maps

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 $\mathfrak{p}$-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.

Keywords:

• Bhargava factorial
• discrete valuation ring
• Stirling formula

2020 Mathematics Subject Classification:

• 13F20
• 11C08
• 11B83

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.

Additional information

Funding

This work was supported by Grant-in-Aid for JSPS Research Fellow (Grant Number: JP19J10705).