On the generating polynomials for the distribution of generalized binomial coefficients in discrete valuation domains

Abstract

For a discrete valuation domain $V$ with maximal ideal $\mathfrak{m}$ such that the residue field $V/\mathfrak{m}$ is finite, there exists a sequence of polynomials $(F_n(x))_{n \ge 0}$ defined over the quotient field $K$ of $V$ that forms a basis of the $V$-module $\text{Int}(V) = \{f \in K[x] | f(V)\subseteq V\}$. This sequence of polynomials bears many resemblances to the classical binomial polynomials $(\binom{x}{n})_{n \ge 0}$. In this paper, we introduce a generating polynomial to account for the distribution of the $V$-values of the polynomials $F_n(x)$ modulo the maximal ideal $\mathfrak{m}$, and prove a result that provides a method for counting exactly how many $V$-values of the polynomials $(F_n(x))_{n \ge 0}$ fall into each of the residue classes modulo $\mathfrak{m}$. Our main theorem in this paper can be viewed as an analogue of the classical theorem of Garfield and Wilf in the context of discrete valuation domains.