Geometrically reductive algebraic group
= 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 $F(v)\ne0$. 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.