  Let $G$ be a linear algebraic group over a field $k$, and let $V$ be a
$G$-module. Recall that the nullcone of $(G,V)$ is the set of points $v$ in $V$
with the property that $f(v)=0$ for every positive degree homogeneous invariant
$f$ in $k[V]^G$. We define numbers $\delta(G,V)$ and $\sigma(G,V)$ associated
with a given representation as follows: $\delta(G,V)$ is the smallest number
$d$ such that, for any point $v$ in $V^G$ outside the nullcone, there exists an
invariant $f$ of degree at most $d$ such that $f(v)$ is not zero; $\sigma(G,V)$
is the same thing with $V^G$ replaced by $V$. If k has positive characteristic,
we show that $\delta(G,V)$ is infinite for all subgroups of $GL_2(k)$
containing a unipotent subgroup, and that $\sigma(G,V)$ is finite if and only
if $G$ is finite. If $k$ has characteristic zero we show that $\delta(G,V)=1$
for all linear algebraic groups and that if $\sigma(G,V)$ is finite then the
connected component of $G$ is unipotent.
