\begin{definition}[Definition:Multiplication of Polynomials/Polynomial Forms]
Let $\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$ and $\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomial forms in the indeterminates $\set {X_j: j \in J}$ over $R$.
The '''product of $f$ and $g$''' is defined as:
:$\ds f \circ g := \sum_{k \mathop \in Z} c_k \mathbf X^k$ 
where:
:$\ds c_k = \sum_{\substack {p + q \mathop = k \\ p, q \mathop \in Z} } a_p b_q$
\end{definition}