Definition:Multiplication of Polynomials/Polynomial Forms
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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.