Definition:Null Polynomial
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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.
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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
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Also known as
The null polynomial can also be referred to as the zero polynomial.