Definition:Multiplication of Polynomials

Definition

Let $\left({R, +, \circ}\right)$ be a ring.

Let $\left({S, +, \circ}\right)$ be a subring of $R$.

Let $x \in R$.

Let:

$\displaystyle f = \sum_{j \mathop = 0}^n a_j x^j$
$\displaystyle g = \sum_{k \mathop = 0}^n b_k x^k$

be polynomials in $x$ over $S$ such that $a_n \ne 0$ and $b_m \ne 0$.

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

$\displaystyle f g := \sum_{l \mathop = 0}^{m + n} c_l x^l$

where:

$\displaystyle \forall l \in \left\{{0, 1, \ldots, m+n}\right\}: c_l = \sum_{\substack{j \mathop + k \mathop = l \\ j, k \mathop \in \Z}} a_j b_k$

Polynomial Forms

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:

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

where:

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

Polynomials as Sequences

Let:

$f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right)$

and:

$g = \left \langle {b_k}\right \rangle = \left({b_0, b_1, b_2, \ldots}\right)$

Then the operation of (polynomial) multiplication is defined as:

$f g := \left({c_0, c_1, c_2, \ldots}\right)$

where $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j b_k$