rational function RationalFunction 2003-05-26 03:46:51 2005-03-27 22:48:06 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \def\sse{\subseteq} \def\bigtimes{\mathop{\mbox{\Huge $\times$}}} \def\impl{\Rightarrow} A real function $R(x)$ of a single variable $x$ is called \emph{\PMlinkescapetext{rational}} if it can be written as a quotient \[ R(x) = \frac{P(x)}{Q(x)}, \] where $P(x)$ and $Q(x)$ are polynomials in $x$ with real coefficients. When one is only interested in algebraic properties of $R(x)$ or $P(x)$ and $Q(x)$, it is convenient to forget that they define functions and simply treat them as algebraic expressions in $x$. In this case $R(x)$ is referred to as a \emph{rational expression}. In general, a rational function (expression) $R(x_1,\ldots,x_n)$ has the form \[ R(x_1,\ldots,x_n) = \frac{P(x_1,\ldots,x_n)}{Q(x_1,\ldots,x_n)}, \] where $P(x_1,\ldots,x_n)$ and $Q(x_1,\ldots,x_n)$ are polynomials in the variables $(x_1,\ldots,x_n)$ with coefficients in some field or ring $S$. In this sense, $R(x_1,\ldots,x_n)$ can be regarded as an element of the fraction field $S(x_1,\ldots,x_n)$ of the polynomial ring $S[x_1,\ldots,x_n]$.