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Homerational function

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# rational function

A real function $R(x)$ of a single variable $x$ is called
*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 *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 $S[x_{1},\ldots,x_{n}]$.

## Mathematics Subject Classification

26C15*no label found*

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