Functional Degrees And Arithmetic Applications, I: The Set Of Functional Degrees

TL;DR

This paper advances the calculus of functional degrees for maps between commutative groups, determining the maximum finite degree and all possible degrees, and applies this to solve a problem related to the nilpotency index of certain group ring ideals.

Contribution

It computes the maximum finite functional degree for maps between fixed commutative groups and characterizes all possible degrees, extending the calculus of functional degrees.

Findings

Identified the largest possible finite functional degree for maps between given commutative groups.

Determined the set of all possible degrees of such maps.

Provided a solution to the nilpotency index problem of augmentation ideals in specific group rings.

Abstract

We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any fixed given commutative groups $A$ and $B$, we compute the largest possible finite functional degree that a map $f: A \longrightarrow B$ can have. We also determine the set of all possible degrees of such maps. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form $Z_{p^\beta}[Z_{p^{\alpha_1}}\times Z_{p^{\alpha_2}}\times\dotsm\times Z_{p^{\alpha_n}}]$ with $p,\beta,n,\alpha_1,\dotsc,\alpha_n\in\mathbb{Z}^+$, $p$ prime.