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 $\family {X_j: j \in J}$ 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:
 * $\ds \map \deg f = \max \set {\size {k_r}: i = 1, \ldots, r}$

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

Some sources denote $\map \deg f$ by $\partial f$.