definiton of homogenous function

Question

according to different sources in the internet, homogeneous function is defined as if $f(tx,ty)=t^mf(x,y)$ for some integer $m$, but my question is why integer, why not a real number(assuming the function is defined over reals to reals), consider this function $f(x,y)=(x^2+y^2)^{1/3}$, this intuitively seems to be a homogeneous function, but according to the definition it is not. What is the reason? and is $f(x,y)=(x^2+y^2)^{1/2}$ a homogeneous function, since $f(tx,ty)=|t|f(x,y)$, so I suppose it shouldn't be a homogeneous function, but please verify it(and are these the reason for it to be defined over integers?)

Answer 1

Related: Is a function still homogeneous if it factors out the absolute value of $t$ or a non-integer exponent of $t$?

It depends on the definition you are using. $m$ does not necessarily need to be an integer. Here they define the exponent as being a number.

$f(x,y)=(x^2+y^2)^{1/3}$ would satisfy this definition as $$f(tx,ty)=((tx)^2+(ty)^2)^{1/3}=t^{2/3}f(x,y)$$ with $m=2/3$.

And, $(x^2+y^2)^{1/2}$ is homogeneous of degree 1. See the related link for more.