Definition:Multiplication of Polynomials/Polynomial Forms

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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 polynomial forms in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

The product of $f$ and $g$ is defined as:

$\displaystyle f \circ g := \sum_{k \mathop \in Z} c_k \mathbf X^k$


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

Also see

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