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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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  • $\begingroup$ This just means the polynomial ring in the variable $x$ with coefficients in $\mathbb{R}$. $\endgroup$ – Adrián Barquero Dec 24 '10 at 17:38
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    $\begingroup$ 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 $a_j \in \mathbb{R}$. $\endgroup$ – Adrián Barquero Dec 24 '10 at 17:40
  • $\begingroup$ In that context, $ [x]$ does not mean nearest integer $\endgroup$ – J. W. Tanner Jul 10 at 15:24
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$\mathbb{R}[x]$ 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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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 $\rm\,s\,.\,$ Equivalently $\,\rm R[s]$ is the image of $\rm\,R[x]\,$ under the evaluation map $\rm\,x\mapsto s,\,$ i.e. elements of $\rm\,S\,$ writable as a polynomial in $\rm\,s\,$ with coefficents in $\rm\,R.\,$

Similarly, if $\rm\,F \subset E\,$ are fields and $\rm\,\alpha\in E\,$ then $\rm\,F(e)\,$ denotes the field-adjunction of $\rm\,\alpha\,$ to $\rm\,F,\,$ i.e. the smallest subfield of $\rm\,E\,$ containing both $\rm\,F\,$ and $\rm\,\alpha.$

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 most general ring obtained by adjoining to $\rm\,R\,$ 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 $\rm\,f(x)= \,$ 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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