Degree of Rational Function

This might sound like a very trivial question but I found different answers on the web.

Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What is the degree of the rational function?

Is it the maximum degree of $f$ and $g$? Or is it $\deg(f) – \deg(g)$?

Thanks

1 Answer

The convention that I have seen is that the degree of the rational function $$s(x) := \frac{f(x)}{g(x)},$$ where $f$ and $g$ are polynomials that have no common factors, is $$\deg s := \max\{\deg f, \deg g\} .$$One motivation for this definition is that, in analogy with the notion of degree of a polynomial, over $\Bbb C$ the equation $$s(x) = w$$ has $\deg s$ solutions (in the Riemann sphere, and counting multiplicity) for generic $w \in \Bbb C$. Indeed, we can rearrange $s(x) = w$ as the polynomial equation $$f(x) - w g(x) = 0$$ and, when $\deg f \neq \deg g$ (and most of the time when $\deg f = \deg g$), the degree of the polynomial $f - w g$ is $\max\{\deg f, \deg g\} = \deg s$.