Definition:Polynomial over Ring/Multiple Variables

Definition
Let $R$ be a commutative ring with unity. Let $I$ be a set.

A polynomial over $R$ in $I$ variables is an element of a polynomial ring in $I$ variables over $R$.

Thus:
 * Let $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial

is a short way of saying:
 * Let $R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\left\langle{X_i}\right\rangle_{i \mathop \in I}$, and let $P$ be an element of this ring.

Also see

 * Definition:Polynomial over Ring in One Variable