Definition:Degree of Polynomial/Polynomial Form

Definition
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{{X_j: j \in J}\right\}$ for some multiindices $k_1, \ldots, k_r$.

Let $f$ not be the null polynomial.

Let $k = \left({k_j}\right)_{j \mathop \in J}$ be a multiindex.

Let $\displaystyle \left|{k}\right| = \sum_{j \mathop \in J} k_j \ge 0$ be the degree of the mononomial $\mathbf X^k$.

The degree of $f$ is the supremum:
 * $\displaystyle \deg \left({f}\right) = \max \left\{{\left| {k_r} \right|: i = 1, \ldots, r}\right\}$

Also known as
The degree of a polynomial form $f$ is also sometimes called the order of $f$.

Some sources denote $\deg \left({f}\right)$ by $\partial f$.