Definition:Null Polynomial

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One Variable

Let $R$ be a commutative ring with unity.

Let $P \in R[X]$ 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 $R[X]$.


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 $f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right)$ 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.


Also see