 Geometrically reductive algebraic group

For every finite-dimensional rational representation $V$ and every nonzero $v\in V^G$, a geometrically reductive algebraic group admits a homogeneous invariant polynomial $F\in k[V{]}^G$ of positive degree with $F(v)\ne 0$. A linearly reductive group can take degree one. In positive characteristic, higher degree is essential: a power $p^r$ can separate a fixed vector even when an invariant linear functional cannot.