Coefficients of Polynomial Product

Theorem
Let $p_1, \ldots p_n$ be polynomial forms in the indeterminates $\left\{{X_j : j \in J}\right\}$ over a commutative ring $R$.

Suppose that for each $i$ with $1 \le i \le n$, we have, for appropriate $a_{i,k} \in R$:


 * $p_i = \displaystyle \sum_{k \mathop \in Z} a_{i,k} X^k$

where $Z$ comprises the multiindices of natural numbers over $J$.

Then:


 * $\displaystyle \prod_{i \mathop = 1}^n p_i = \displaystyle \sum_{k \mathop \in Z} b_k X^k$

where:


 * $\displaystyle b_k := \sum_{k_1 + \cdots + k_n = k} \left({\prod_{i \mathop = 1}^n a_{i,k_i} }\right)$