Let $X\subseteq\Bbb C^n$ be a smooth affine variety and $G$ a complex reductive group acting algebraically on $X$. 


Let $x\_0\in X$. If there is a non-constant $G$-invariant *holomorphic* function $f:X\to\Bbb C$ such that $f(x\_0)\neq 0$, must there also exist a non-constant $G$-invariant *polynomial* $p\in\Bbb C[X]$ such that $p(x\_0)\neq 0$?



$G$-invariant holomorphic vs. polynomial functions
**Lemma** (Dolgachev's Lectures on Invariant Theory). Let $X$ be an affine $G$-variety, and let $Z\_1$ and $Z\_2$ be two closed $G$-invariant
subsets with $Z\_1\cap Z\_2 = \emptyset$. Assume $G$ is geometrically reductive. Then there exists a $G$-invariant function $\varphi \in \mathcal{O}(X)^G$ such that $\varphi(Z\_1) = 0$, $\varphi(Z\_2) = 1$.


If $X\not=\overline{Gx\_0}$, then letting $Z\_2=\overline{Gx\_0}$ and $Z\_1$ any closed orbit in $X-\overline{Gx\_0}$ shows that non-constant $G$-invariant polynomials that are not 0 at $x\_0$ always exist. Since you assumed $X$ is smooth, these will be holomorphic too.


