Suppose that a finite group acts on two vector spaces $X$ and $Y$, and that $f:X\longrightarrow Y$ is an equivariant polynomial map which is homogeneous of degree $n$, and that there does not exist any nonzero equivariant polynomial map $X\longrightarrow Y$ with smaller degree.

What does $f^{-1}(0)$ look like?

Can you think of an example in which $f^{-1}(0)$ is not a finite union of linear subspaces?

For what groups must $f^{-1}(0)$ be a finite union of linear subspaces? In what situations must $f^{-1}(0)$ be nice? (For example, maybe for some groups, if the action on $Y$ is irreducible, $f^{-1}(0)$ a is finite union of submanifolds).

I am interested in the case where $X$ and $Y$ are real vector spaces, but if you know the answer for complex vector spaces, I would like to know that too.

The slightly more difficult question I am really interested in is the following: Given a generic equivariant smooth map $f:X\longrightarrow Y$, what does $f^{-1}(0)$ look like? If it is a (locally) finite union of submanifolds, then I would be surprised and happy.