What is a "rational function of $n$ variables"?

Question

The only definition of a rational function I was able to find is that of Varsity Tutors.

"A rational function is defined as the quotient of polynomials in which the denominator has a degree of at least $1$"

If we are talking merely about $x$, then I get the concept. A rational function $f(x)$ could be written as "$\frac{p(x)}{q(x)}$, where $q(x) \neq 0$."

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The issue that I'm having is that of talking about rational functions of $n$ variables. For instance, what would be the meaning of '$f(x,y)$ is a rational function of $x$ & $y$' ?

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I would truly appreciate any help/thoughts

Answer 1

For instance, $f(x,y)=\frac {p (x,y)}{q (x,y)} $ is a rational function of $x $ and $y $ if $p $ and $q $ are polynomials in $x $ and $y $... So $p $ and $q $ are finite sums of terms of the form $ax^ny^m $, where $a $ is a numerical coefficient and $n $ and $m $ are whole numbers. .. and, of course, the degree of $q $ is $1$ or more. ..

Answer 2

The definition applies to $n$ variables as well. For any two polynomials $p,q$, the function $f(x_1,x_2,\dots x_n)=\frac{p(x_1,x_2,\dots x_n)}{q(x_1,x_2,\dots x_n)}$ is a rational function (if $q$ is not constant).

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For example, $f(x,y)=\frac xy$ is a rational function of $2$ variables.