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 ${\left(\left(\genfrac{}{}{0.0pt}{}{x}{n}\right)\right)}_{n\ge 0}$. 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.