Spring 2023 GENERATING POLYNOMIALS FOR THE DISTRIBUTION OF GENERALIZED BINOMIAL COEFFICIENTS IN DISCRETE VALUATION DOMAINS
Dong Quan Ngoc Nguyen
J. Commut. Algebra 15(1): 65-73 (Spring 2023). DOI: 10.1216/jca.2023.15.65

Abstract

For a discrete valuation domain V with maximal ideal 𝔪 such that the residue field V𝔪 is finite, there exists a sequence of polynomials (Fn(x))n0 defined over the quotient field K of V that forms a basis of the V-module Int(V)={fK[x]|f(V)V}. This sequence of polynomials bears many resemblances to the classical binomial polynomials (xn)n0. In this paper, we introduce a generating polynomial to account for the distribution of the V-values of the polynomials Fn(x) modulo the maximal ideal 𝔪, and prove a result that provides a method for counting exactly how many V-values of the polynomials (Fn(x))n0 fall into each of the residue classes modulo 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.

Citation

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Dong Quan Ngoc Nguyen. "GENERATING POLYNOMIALS FOR THE DISTRIBUTION OF GENERALIZED BINOMIAL COEFFICIENTS IN DISCRETE VALUATION DOMAINS." J. Commut. Algebra 15 (1) 65 - 73, Spring 2023. https://doi.org/10.1216/jca.2023.15.65

Information

Received: 3 August 2020; Revised: 13 April 2022; Accepted: 15 April 2022; Published: Spring 2023
First available in Project Euclid: 20 June 2023

MathSciNet: MR4604786
zbMATH: 07725175
Digital Object Identifier: 10.1216/jca.2023.15.65

Subjects:
Primary: 11B65 , 13A18 , 13F20

Keywords: discrete valuation domain , generalized binomial coefficient , Lucas’ theorem , very well distributed and well-order sequence

Rights: Copyright © 2023 Rocky Mountain Mathematics Consortium

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Vol.15 • No. 1 • Spring 2023
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