\begin{definition}[Definition:Addition of Polynomials/Polynomial Forms]
Let:
:$\ds f = \sum_{k \mathop \in Z} a_k \mathbf X^k$
:$\ds g = \sum_{k \mathop \in Z} b_k \mathbf X^k$
be polynomials in the indeterminates $\set {X_j: j \in J}$ over $R$.
{{explain|What is $Z$ in the above? Presumably the integers, in which case they need to be denoted $\Z$ and limited in domain to non-negative? However, because $Z$ is used elsewhere in the exposition of polynomials to mean something else (I will need to hunt around to find out exactly what), I can not take this assumption for granted.}}
The operation '''polynomial addition''' is defined as:
:$\ds f + g := \sum_{k \mathop \in Z} \paren {a_k + b_k} \mathbf X^k$
The expression $f + g$ is known as the '''sum''' of $f$ and $g$.
\end{definition}