Conjugacy classes and invariant subrings of R-automorphisms of R[x]

TL;DR

This paper investigates the structure of automorphism groups of polynomial rings over rings of integers modulo n, focusing on conjugacy classes and invariant subrings, with explicit results for the case n=4.

Contribution

It characterizes conjugacy classes in R-automorphisms of R[x] and explicitly describes invariant subrings for the case n=4.

Findings

Conjugacy classes in the automorphism group G are classified.

Explicit structure of G is determined for n=4.

All invariant subrings under subgroups of G are described for n=4.

Abstract

We consider the group G of R-automorphisms of the polynomial ring R[x] especially in the case where R is the ring of integers modulo n. We describe conjugacy classes in G, and in the case where n = 4, we describe more explicitly the structure of G and determine all rings of invariants of R[x] with respect to subgroups of G.