In commutative algebra with many variables, repeating lists of variables in polynomial arguments and various rings gets very tedious. I suggest using $X_{1..n}$ instead of $X_1,\ldots,X_n$.

Here's an excerpt from Bourbaki's Commutative Algebra, page 222:

For every formal power series $f\in A[[X_1,\ldots,X_n]]$, $$f(X_1,\ldots,X_n)-f(Y_1,\ldots,Y_n)=\sum_{i=1}^n (X_i-Y_i)h_i(X_1,\ldots,X_n,Y_1,\ldots,Y_n)$$ where the $h_i$ belong to $A[[X_1,\ldots,X_n,Y_1,\ldots,Y_n]]$.

And here's how it looks with my suggested notation:

For every formal power series $f \in A[[X_{1..n}]]$, $$ f(X_{1..n})-f(Y_{1..n})=\sum_{i=1}^n (x_i-Y_i)h_i(X_{1..n},Y_{1..n})$$ where the $h_i$ belong to $A[[X_{1..n},Y_{1..n}]]$.

I think this notation is maximally succinct, and helps a reader from getting lost in long lists of variables.