# Definition:Null Polynomial

## One Variable

Let $R$ be a commutative ring with unity.

Let $P \in R \sqbrk X$ be a polynomial over $R$.

### Definition 1

The polynomial $P$ is a **zero polynomial** if and only if it is the zero element of the polynomial ring $R \sqbrk X$.

### Definition 2

The polynomial $P$ is a **zero polynomial** if and only if all its coefficients are zero.

### Ring

Let $\left({R, +, \times}\right)$ be a ring.

The zero $0_R$ of $R$ can be considered as being the **null polynomial over $R$** of any arbitrary element $x$ of $R$.

### Polynomial Form

Let $f = a_1 \mathbf X^{k_1} + \cdots + a_r \mathbf X^{k_r}$ be a polynomial form over $R$ in the indeterminates $\left\{{X_j: j \in J}\right\}$.

For all $i = 1, 2, \ldots, r$, let $a_i = 0$.

Then $f$ is the **null polynomial** in the indeterminates $\left\{{X_j: j \in J}\right\}$.

### Sequence

Let $f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}$ be a polynomial over a field $F$.

Let $0_F$ be the zero of $F$.

Let $a_0 = a_1 = a_2 = \ldots = 0_F$.

Then $f$ is known as the **null polynomial**.

## Multiple Variables

## Also known as

The **null polynomial** can also be referred to as the **zero polynomial**.