If a mapgerm f :Rn, 0 → Rp, 0
is a submersion (rkf = p), then its zero set is regular (the germ of a manifold)
by the Implicit Function Theorem. Of course, there are also critical maps
(rkf < p) whose zero sets are manifolds. Submersions have the added feature
that one can discern that the zero set is regular from the first derivative
of f at 0. Are there other instances in which one can tell purely from the
derivatives of f at 0 that the zero set is regular? In this paper we show that
there are, and go part way toward the eventual goal of describing them
all.
