\begin{definition}[Definition:Degree of Polynomial/Polynomial Form]
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\family {X_j: j \in J}$ for some multiindices $k_1, \ldots, k_r$.
Let $f$ '''not''' be the null polynomial.
Let $k = \family {k_j}_{j \mathop \in J}$ be a multiindex.
Let $\ds \size k = \sum_{j \mathop \in J} k_j \ge 0$ be the degree of the monomial $\mathbf X^k$.
The '''degree of $f$''' is the supremum:
:$\ds \map \deg f = \max \set {\size {k_r}: i = 1, \ldots, r}$
\end{definition}