  Gotzmann's Persistence states that the growth of an arbitrary ideal can be
controlled by comparing it to the growth of the lexicographic ideal. This is
used, for instance, in finding equations which cut out the Hilbert scheme (of
subschemes of $\mathbf{P}^n$ with fixed Hilbert polynomial) sitting inside an
appropriate Grassmannian. We introduce the notion of an {\it extremal ideal}
which extends the notion of the lex ideal to other term orders. We then state
and prove a version of Gotzmann's theorem for these ideals, valid in an open
subset of a Grassmannian.
