If $G$ is a finite subgroup of $GL_n(C)$ then there is a polynomial $f\in C[x_1,...,x_n]$ such that $f(gv)=f(v)$

Probably the easiest construction is as follow: Let $h(v)$ be any non-trivial polynomial, and then let

$$f(v) = \prod_{g \in G} h(gv).$$

It's both non-constant and invariant by $G$. If $h$ was homogenous of degree one, then $f$ is homogenous of degree $|G|$.

More generally, if $V$ is the representation of $G$ on the linear terms, then one can study the ring of $G$-invariants $k[V]^{G}$. Hilbert proved that this ring is always finitely generated. It's also easy to show it has transcendence degree $n$. This goes under the name "invariant theory".