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Suppose:

  1. $cf(x,y)=f(cx,cy)$
  2. $f(x,y)=f(y,x)$

If $f$ is a polynomial, then $f(x,y)=c(x+y)$ because by Euler's homogeneous function theorem, $f(x,y)=xf_x(x,y)+yf_y(x,y)$ where $f_x,f_y$ are homogeneous of degree zero, and constants are the only such polynomials.

It seems like maybe the result holds more generally (i.e. even if we don't know a priori that $f$ is a polynomial) but I'm having difficulty finding counterexamples or proving it either way.

One thing I noticed is if $f'$ is homogenous of degree zero then we can parameterize it in polar coordinates using just the angle (since magnitude can't affect the outcome, by definition). So we can rewrite this as:

$$f(x,y)=xg_x(\theta)+yg_y(\theta)=xg_x(\tan^{-1}(y/x))+yg_y(\tan^{-1}(y/x))$$

But now here's where I start to get confused. $tan^{-1}(y/x)$ is clearly not a symmetric function. So does that mean that I've proven the $g$'s must be constant? The g's don't have to be symmetric, but it's hard to come up with an example which works with non-constant g's.

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1 Answer 1

up vote 2 down vote accepted

Let $f(x,y)=\dfrac{x^3+y^3}{x^2+y^2}\,$ if $\,(x,y)\ne (0,0)$ and let $f(0,0)=0$.

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Good point. I guess more generally $f=g/h$ with g having one degree higher than h will work, so there's nothing really special we can say about homogeneous functions of degree 1. –  Xodarap Sep 17 '12 at 18:22
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