You has a rational function f(x)g(x), where f(x) and g(x) are polynomials. What is the degree of the rational function? This might sound like a very trivial question but I found different answers on the web. Is it the maximum degree of f and g? Or is it deg(f)−deg(g)?

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You has a rational function   f(x)g(x),   where f(x) and g(x) are polynomials. What is the degree of the rational function?   This might sound like a very trivial question but I found different answers on the web.   Is it the maximum degree of f and g? Or is it deg(f)−deg(g)?

Derrick Woods · Accepted Answer

The convention that I have seen is that the degree of the rational function s(x):=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 CC the equation s(x)=w has deg⁡s solutions (in the Riemann sphere, and counting multiplicity) for generic w∈CC. Indeed, we can rearrange s(x)=w as the polynomial equation f(x)−wg(x)=0 and, when deg⁡f≠deg⁡g (and most of the time when deg⁡f=deg⁡g), the degree of the polynomial f−wg is max{deg⁡f,deg⁡g}=deg⁡s.

You has a rational function \frac{f(x)}{g(x)}, where f(x) and g(x) are polynomials.

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