Classification of Polynomial Mappings Between Commutative Groups

TL;DR

This paper characterizes and classifies polynomials with rational coefficients that induce well-defined maps between finite commutative groups, providing interpolation formulas and a Taylor-type theorem for their computation.

Contribution

It offers a complete characterization and classification of such polynomials and maps, along with explicit formulas for their calculation.

Findings

Characterization of polynomials inducing maps between cyclic groups

Classification of all polynomial-induced maps between finite commutative groups

Interpolation formulas and Taylor-type theorem for polynomial calculation

Abstract

Some polynomials $P$ with rational coefficients give rise to well defined maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto P(x)+r\Z$. More generally, there are polynomials in several variables with tuples of rational numbers as coefficients that induce maps between commutative groups. We characterize the polynomials with this property, and classify all maps between two given finite commutative groups that arise in this way. We also provide interpolation formulas and a Taylor-type theorem for the calculation of polynomials that describe given maps.