How to write an arbitrary polynomial in $n$ variables

Question

Be $k$ a field. I'm trying to define a function on $k[x_1, ..., x_n]$. However, I know of no way to write an arbitrary element of this ring efficiently. I read somewhere about using the $S_n$-orbit, but that seemed cumbersome too. Maybe I'll just have to accept it is cumbersome, but any ideas are appreciated.

Answer 1

What $k[x_1,\dots,x_n]$ actually means is the free commutative $k$-algebra generated from the set $\{x_1,\dots,x_n\}$. We can generalize this to $k[I]$ for any set $I$. Of course, just writing $k[I]$ without comment will definitely lead to confusion. If I were writing a sizable article that leverages this notation a bit, I may well define the notation for an arbitrary set, and then state that $k[x_1,\dots,x_n]$ is shorthand for $k[\{x_1,\dots,x_n\}]$ (assuming all $x_i$ are distinct).¹

For your purposes, a more practical compromise would be to write $k[\{x_i\}_{i\in I}]$ where you also have $\{f_i\}_{i\in I}$ and $\rho$ becomes simply $\rho(x_i)(v)=f_i(v)$. Often $[n]$ (or just $n$ itself) is used for notation for the $n$-element set $\{0,\dots,n-1\}$ (or $\{1,\dots,n\}$ if you're uncouth), so you could say $k[\{x_i\}_{i\in[n]}]$ and $\{f_i\}_{i\in[n]}$.

Notation like $\{a_i\}_{i\in I}$ is commonly used to represent the $I$-indexed family of objects $a_i$. If all the $a_i$ are drawn from some encompassing set, $A$, this can be identified with a function $I\to A$. In set theory, such a function is a set of pairs, so if you wanted to be really precise, you should have something like $\rho((i,x_i))(v)=f_i(v)$.

¹ Actually, knowing me, I would not do this, and would only mention this to explain the connection to more typical notation if I mentioned it at all.