Let $f: (x_1,\ldots,x_n) \mapsto (F_1,\ldots,F_n)$ be a $k$-algebra endomorphism of $k[x_1,\ldots,x_n]$, where $F$ is a field and $n \geq 1$.
When one says that, after a change of variables, we can assume that $F_1,\ldots,F_n$ has a certain property, does one mean the following:
There exists a $k$-algebra automorphism $g$ of $k[x_1,\ldots,x_n]$, such that $g(F_1),\ldots,g(F_n)$ has that certain property.
Please, is this correct?
I guess that my question is quite trivial, but I wish to be sure that I am not missing something. More precisely, I wish to be sure that we are talking about $g(F_i)=g(f(x_i))=(gf)(x_i)$ and not about $(fg)(x_i)$.
Thank you very much!
This phrase is ambiguous and could have several meanings: it could refer to a composition of the form $gf$, or it could refer instead to $fg$, or it could refer to the conjugation $gfg^{-1}$, or it might even refer to something like $gfh$ where $g$ and $h$ are two automorphisms. You'll have to figure out from context what meaning is intended (it should usually be clear in context). Lacking any context, I would guess that the conjugation $gfg^{-1}$ is the most likely meaning.