Does anyone know any result on finitely determined germs to help me prove that the germ $f(x,y)=x^3+ xy^3$ is $4$- determined? I tried using the definition of germs finitely determined, which is:$f: \mathbb{R}^n \rightarrow \mathbb{R}$ is $k$-determined if for any other germ $g: \mathbb{R}^n \rightarrow \mathbb{R}$ such that the $k$-jet of g is equal to $k$-jet of the $f$, then $f$ and $g$ are right equivalents, i. e., exist a difeomorfism $h$ such that $f=g\circ h$, but not getting success. I think there should be some results to help me prove it. Thanks!
germ finitely determined
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differential-geometry
germs
singularity-theory