Definition:Multiplication of Polynomials/Polynomial Forms

Definition
Let $\displaystyle f = \sum_{k \mathop \in Z} a_k \mathbf X^k$, $\displaystyle g = \sum_{k \mathop \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

We define the product:
 * $\displaystyle f \circ g := \sum_{k \mathop \in Z} c_k \mathbf X^k$

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

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