Two-Variable Polynomials

Question

Let $f(x,y) = ax^2 +bxy +cy^2$, then: $$ f(tx,ty)=t^2ax^2+t^2bxy+t^2cy^2=t^2f(x,y) $$ Now, if there is a function $g(x,y)$ such that $g(tx,ty)=t^2g(x,y)$, can we conclude that $g(x,y)$ has the form $ax^2+bxy+cy^2$? If not, what conditions do we need to make such a conclusion?

Answer 1

$g(x,y)=x^{2}\sin(\frac y x)$ if $x \neq 0$, $g(x,y)=0$ if $x=0$ is a counterexample.