The Largest Possible Finite Degree of Functions between Commutative Groups

TL;DR

This paper characterizes functions of finite degree between commutative groups as polyfracts and determines the maximum finite degree possible, solving a problem related to the nilpotency of certain group rings.

Contribution

It introduces polyfracts as a characterization of finite degree maps and calculates the maximal finite degree between finite commutative groups, addressing an open problem.

Findings

Finite degree maps are exactly polyfracts.

The degree of a polyfract equals its functional degree.

Determined the maximum finite degree between two finite commutative groups.

Abstract

We consider maps between commutative groups and their functional degrees. These degrees are defined based on a simple idea -- the functional degree should decrease if a discrete derivative is taken. We show that the maps of finite functional degree are precisely the maps that can be written as polyfracts, as polynomials in several variables but with binomial functions in the place of powers. Moreover, the degree of a polyfract coincides with its functional degree. We use this to determine the largest possible finite functional degree that the maps between two given finite commutative groups can have. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency degree of the augmentation ideal of the group ring $Z_{p^\beta}[Z_{p^{\alpha_1}}\times Z_{p^{\alpha_2}}\times\dots\times Z_{p^{\alpha_n}}]$. Some generalizations and simplifications of proofs to underlying facts are presented, too.