Quasi-homogeneous smooth functions vs. polynomials
Andrew Mclaughlin 
I am dealing with the following problem.
Assume for simplicity that we are dealing with function of two variables only $f = f(x,y)$. Let $|x|,|y| \in \mathbb{Z}$ be two non-zero integers, called weights of the coordinates $(x,y)$. One says that $f$ is quasi-homogeneous of degree $\lambda \in \mathbb{Z}$, if for all $t > 0$, one has $$ f(t^{|x|}x, t^{|y|}y) = t^{\lambda} f(x,y) $$
For $|x| = |y| = 1$, one obtains usual homogeneous function of degree $\lambda$.
Classic example of such function is a polynomial in $(x,y)$ of a total weight $\lambda$, where by "total weight" of the polynomial one means the weighted sum of the corresponding powers, for example $$ |x^{2}y^{3}| = 2 |x| + 3 |y| $$
Now the question:
Assume that $f$ is quasi-homogeneous and smooth on $\mathbb{R}^{2}$. Is every such function a polynomial of the total weight $\lambda$? Of course not necessarily a monomial.
Comments on solution:This is true for homogeneous functions, that is $|x| = |y| = 1$. The usual proof uses the fact that partial derivatives are homogeneous functions of lower degree. In the more general case, one can show that $$ (\partial_{x}f)(t^{|x|}x, t^{|y|}y) = t^{\lambda-|x|} (\partial_{x}f)(x,y)), $$ $$ (\partial_{y}f)(t^{|x|}x, t^{|y|}y) = t^{\lambda-|y|} (\partial_{y}f)(x,y), $$ for all $t > 0$. One can iterate this to find similar expressions for any higher partial derivatives. If $|x|, |y| > 0$, one can show easily that any quasi-homogeneous continuous function of negative degree has to be $0$. Starting with $\lambda \geq 0$, one uses this observation to find that at certain point all (higher) partial derivatives of $f$ vanish, as the degree becomes negative for all of them. This already implies that $f$ has to be a polynomial. Showing that it has to have a total weight $\lambda$ is easy.
If both weights are strictly negative, $|x|, |y| < 0$, the proof is analogous, except in this case all quasi-homogeneous functions of strictly positive degree are identically zero.
I have problems to show it (or find counterexample) for e.g $|x| > 0$ and $|y| < 0$. Consider for example $|x| = 1$ and $|y| = -1$. Then $f(x,y) = x^{n} y^{n}$ is a quasi-homogenous smooth function (and polynomial) of degree $0$. Clearly there is now no limit on number of partial derivatives, so I must find another way to characterize polynomials and the simple analogue of the above proof cannot work....
Does anyone have any ideas? Thank you in advance.
$\endgroup$1 Answer
$\begingroup$You can compose quasi-homogeneous function of degree 0 with any smooth function. This will give you plenty of counterexamples such as $\sin(xy)$ or $\sqrt{xy}$ for $|x| = 1, |y| = -1$. Multiplying such functions with appropriate polynomials will provide counterexamples for nonzero $\lambda$.
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