论文标题
在生成多项式上，用于在离散估值域中分布的广义二项式系数
On the generating polynomials for the distribution of generalized binomial coefficients in discrete valuation domains
论文作者
论文摘要
对于带有最大理想$ \ mathfrak {m} $的离散估值域$ v $，使残基$ v/\ mathfrak {m} $是有限的，存在一系列多项式$（f_n（x）_ {n \ ge 0} $ a $ k $ k $ k $ k $ v $ v $ v $ v $ v $ v $ v $ V $ \ text {int}（v）= \ {f \ in K [x] | f（v）\ subseteq v \} $。这种多项式序列与经典的二项式多项式$（\ binom {x} {n}）_ {n \ ge 0} $具有许多相似之处。在本文中，我们介绍了一个生成多项式，以说明多项式的$ v $值$ f_n（x）$ modulo的最大理想$ \ mathfrak {m mathfrak {m} $，并证明一种结果，可以准确地计算polynomials $ ge ge ge ge ge ge ge ge（x）的方法（x）残基类Modulo $ \ Mathfrak {M} $。本文中我们的主要定理可以被视为在离散估值域的背景下，Garfield和Wilf经典定理的类似物。
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.
