Definition:Multiplication of Polynomials/Polynomial Forms

Definition
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:
 * $\ds c_k = \sum_{\substack {p + q \mathop = k \\ p, q \mathop \in Z} } a_p b_q$

Also see
It follows from Polynomials Closed under Ring Product that $f \circ g$ is a polynomial form.