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I ran into this expression in a paper I was reading, and I'm confused about part of the meaning. Here $u$ and $v$ are two polynomials.

$$u, v \in \mathbb{R}[x]$$

I'm not really familiar with usage of $[x]$ here, but if it means "nearest integer", then isn't this expression equivalent to simply:

$$u, v \in \mathbb{Z}$$

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This just means the polynomial ring in the variable $x$ with coefficients in $\mathbb{R}$. – Adrián Barquero Dec 24 '10 at 17:38
When you write $u \in \mathbb{R}[x]$ it just means that $u = a_n x^n + \dots + a_1 x + a_0 $ where the coefficients $a_j \in \mathbb{R}$. – Adrián Barquero Dec 24 '10 at 17:40
up vote 7 down vote accepted

$\mathbb{R}[x]$ is the set of polynomials (with variable $x$) whose coefficients are taken from $\mathbb{R}$, the set of real numbers. It has got nothing to do with the greatest integer function.

Similarly, people talk about polynomials in $\mathbb{Q}[x]$, where coefficients are rational, or more generally, when the coefficients are taken from an arbitrary Ring.

For instance of a usage:

Definition: A real number $r$ is transcendental if and only if, for every $P \in \mathbb{Q}[x]$, we have that $P(r) \neq 0$.

You can find more information here: Polynomial Ring.

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More generally, if $\rm\:R \subset S\:$ are rings and $\rm\:s\in S\:$ then $\rm\:R[s]\:$ denotes the ring-adjunction of $\rm\:s\:$ to $\rm\:R\:,\:$ i.e. the smallest subring of $\rm\:S\:$ containing both $\rm\:R\:$ and $\rm\:s\:.\:$ The notation for the polynomial ring $\rm\:R[x]\:$ is the special case where $\rm\:x\:$ is transcendental over $\rm\:R\ $ (an "indeterminate" in old-fashioned language),$\ $ i.e. $\rm\: x\:$ isn't a root of any polynomial with coefficients in $\rm\:R\:$. One may view $\rm\:R[x]\:$ as the adjunction of a universal (or generic) element $\rm\:x\:$, in the sense that any other adjunction $\rm\:R[s]\:$ is a ring-image of $\rm\:R[x]\:$ under the evaluation homomomorphism $\rm\: x\to s\:.\ $ For example, if $\rm\:R \subset S\:$ are fields then $\rm\:R[s]\cong R[x]/(f(x))\:$ where $\rm\:f(x)\:$ is the minimal polynomial of $\rm\:s\:$ over $\rm\:R\:.\:$ Essentially this serves to faithfully ring-theoretically model $\rm\:s\:$ as a "generic" root $\rm\:x\:$ of the minimal polynomial $\rm\:f(x)\:$ for $\rm\:s\:.\:$ Polynomial rings may be characterized by the existence and uniqueness of such evaluation maps ("universal mapping property"), e.g. see any textbook on Universal Algebra, e.g. Bergman.

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