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Let $K$ be a field. According to Grillet's Abstract Algebra (Second Edition) on page 252,

Every nonzero $\frac{f}{g} \in K(X_1,\,\dots,\,X_n)$ can be written uniquely in the form $\frac{f}{g} = X_1^{m_1} X_2^{m_2} \cdots X_n^{m_n}(h/k)$ with $m_1,\,\ldots,\,m_n \geq 0$ and $k(0,\,\dots,\,0),\,h(0,\,\dots,\,0)\neq 0$;

but I apparently don't understand how this is meant. $k(0,\,\dots,\,0),\,h(0,\,\dots,\,0)\neq 0$ means that the constant part is nonzero, and that seems to be where my understanding ends. For instance, take $n=4$. Then, how is this supposed to work for

$$ \frac{X_1 + X_2}{X_3 + X_4}\quad \text{?} $$

It must be very simple, but I don't get the concept right now or can't read properly for the moment.

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The statement is wrong, as your example shows. The correct statement should be that neither $f$ nor $g$ is divisible by any of the variables $X_1,\cdots,X_n$ (which, when $n=1$, is equivalent to having no constant term, i.e. $f(0),g(0)\ne 0$, which is presumably what the author mixed up with $n>1$).

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