Invariants and conjugacy classes of triangular polynomial maps

In this article, we classify invariants and conjugacy classes of triangular polynomial maps. We make these classifications in dimension 2 over domains containing $\text{Q}$, 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 $(x_1+f_1,\dots ,x_n+f_n)$ where $f_i\in x_n^{p-1}k[x_{i+1}^p,\dots ,x_n^p]$ if $1\le i\le n-1$ and $f_n\in k^*$.