In the older literature they use the term integral rational function. I did google search, math/SE search etc and see that mostly integral rational function is interpreted to mean "polynomial".
But when we read in the older literature that the integral rational function $f=f(x_1,x_2,...,x_n)$ is required to be homogeneous of degree zero in the $x_k$'s, shouldn't it have the form
$$f=\frac{x_9}{x_{15}}$$ or $$f=\frac{x_1 x_5}{x_2 x_7}+\left( \frac{x_1 x_2 x_3}{x_4 x_5 x_6} \right)^3$$
for example? It seems the only way it is homogeneous of degree zero is to have fractions appear in $f$.
The problem I see is that a polynomial by definition does not have variables in the denominator.
Is then an integral rational function always a polynomial? What are the properties of integral rational functions.