Definition:Polynomial

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Real Numbers

A polynomial (in $\R$) is an expression of the form:

$\displaystyle \map P x = \sum_{j \mathop = 0}^n \paren {a_j x^j} = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n - 1} x^{n - 1} + a_n x^n$

where:

$x \in \R$
$a_0, \ldots a_n \in \mathbb k$ where $\mathbb k$ is one of the standard number sets $\Z, \Q, \R$.


Complex Numbers

A polynomial (in $\C$) is an expression of the form:

$\displaystyle P \left({z}\right) = \sum_{j \mathop = 0}^n \left({a_j z^j}\right) = a_0 + a_1 z + a_2 z^2 + \cdots + a_{n-1} z^{n-1} + a_n z^n$

where:

$z \in \C$
$a_0, \ldots a_n \in \mathbb k$ where $\mathbb k$ is one of the standard number sets $\Z, \Q, \R, \C$.


Arbitrary Ring

Let $R$ be a commutative ring with unity.


One Variable

A polynomial over $R$ in one variable is an element of a polynomial ring in one variable over $R$.


Thus:

Let $P \in R \left[{X}\right]$ be a polynomial

is a short way of saying:

Let $R \left[{X}\right]$ be a polynomial ring in one variable over $R$, call its variable $X$, and let $P$ be an element of this ring.


Multiple Variables

Let $I$ be a set.


A polynomial over $I$ in one variable is an element of a polynomial ring in $I$ variables over $R$.

Thus:

Let $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial

is a short way of saying:

Let $R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\left\langle{X_i}\right\rangle_{i \mathop \in I}$, and let $P$ be an element of this ring.


Also see