  We study the group invariant continuous polynomials on a Banach space $X$
that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that
if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L} (X)$, $K$
is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be
separated from $K$ by a continuous polynomial $Q$, then $z$ can also be
separated from $K$ by a $G$-invariant continuous polynomial $P$. It turns out
that this result does not hold when $X$ is a complex Banach space, so we
present some additional conditions to get analogous results for the complex
case. We also obtain separation theorems under the assumption that $X$ has a
Schauder basis which give applications to several classical groups. In this
case, we obtain characterizations of points which can be separated by a group
invariant polynomial from the closed unit ball.
