Invariants and conjugacy classes of triangular polynomial maps

TL;DR

This paper classifies invariants and conjugacy classes of triangular polynomial maps across various dimensions and characteristics, providing a comprehensive understanding of their structure and maximal order maps.

Contribution

It offers a complete classification of invariants and conjugacy classes for triangular polynomial maps in multiple settings, including characteristic p and characteristic zero.

Findings

Classified invariants and conjugacy classes in dimension 2 over domains with Q

Determined invariants and classes of maximal order maps in all dimensions over fields of characteristic p

Identified forms of strictly triangular maps of maximal order in various settings

Abstract

In this article, we classify invariants and conjugacy classes of triangular polynomial maps. We make these classifications in dimension 2 over domains containing $\Q$, dimension 2 over fields of characteristic $p$, and dimension 3 over fields of characteristic zero. We discuss the generic characteristic 0 case. We determine the invariants and conjugacy classes of strictly triangular maps of maximal order in all dimensions over fields of characteristic $p$. They turn out to be equivalent to a map of the form $(x_1+f_1,\ldots,x_n+f_n)$ where $f_i\in x_n^{p-1}k[x_{i+1}^p,\ldots,x_n^p]$ if $1\leq i\leq n-1$ and $f_n\in k^*$.