# Case of Integral rational function

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.

• The answers to your two questions are no, and too long to answer for a single question. – Don Thousand Sep 12 '19 at 15:38
• Which literature are you referring to? – Chappers Sep 12 '19 at 15:39
• Don Thousand, the two examples I gave, are they correct examples of homogeneous integral rational functions of degree 0? – user142523 Sep 12 '19 at 17:08
• Chappers, older literature that refers to homogeneous integral rational functions,, the American Journal of Math from the early 1900s has articles that use this term – user142523 Sep 12 '19 at 17:16
• – Chappers Sep 12 '19 at 17:26