Let $G$ be a finite matrix group, $G \subseteq GL_{n}(\mathbb{C})$. Consider the polynomial ring in $n$ variables; $\mathbb{C}[x_1,...,x_n]$.
It is known that the ivnariant subring $\mathbb{C}[x_1,...,x_n]^G$ is finitely generated.
When looking at generators for this subring, everywhere I read says it is enough to consider the homogeneous polynomials, which are polynomials whose non-zero terms all have the same degree.
Why is this so? Why is it enough to consider these homogeneous polynomials?