Definition:Degree of Polynomial

Definition
Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial in the indeterminates $\left\{{X_j: j \in J}\right\}$ that is not the null polynomial for some multiindices $k_1, \ldots, k_r$.

For a multiindex $k = \left({k_j}\right)_{j \in J}$, let $\displaystyle \left|{k}\right| = \sum_{j \in J} k_j \ge 0$ be the degree of the mononomial $\mathbf X^k$.

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

Some sources write $\deg \left({f}\right)$ as $\partial f$.

The null polynomial is sometimes defined to have degree $-\infty$, but is left undefined in many sources.