On Eisenstein polynomials and zeta polynomials

TL;DR

This paper explores new properties of Eisenstein polynomials and zeta polynomials, drawing analogies to Eisenstein series, and provides finite analogies of their classical properties.

Contribution

It introduces novel analogous properties of Eisenstein polynomials and zeta polynomials, extending the conceptual framework from Eisenstein series.

Findings

New properties of Eisenstein polynomials established

Finite analogies of Eisenstein series properties demonstrated

Enhanced understanding of polynomial analogues in number theory

Abstract

Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.