  Using Green's hyperplane restriction theorem, we prove that the rank of a
Hermitian form on the space of holomorphic polynomials is bounded by a constant
depending only on the maximum rank of the form restricted to affine manifolds.
As an application we prove a rigidity theorem for CR mappings between
hyperquadrics in the spirit of the results of Baouendi-Huang and
Baouendi-Ebenfelt-Huang. Given a real-analytic CR mapping of a hyperquadric
(not equivalent to a sphere) to another hyperquadric $Q(A,B)$, either the image
of the mapping is contained in a complex affine subspace, or $A$ is bounded by
a constant depending only on $B$. Finally, we prove a stability result about
existence of nontrivial CR mappings of hyperquadrics. That is, as long as both
$A$ and $B$ are sufficiently large and comparable, then there exist CR mappings
whose image is not contained in a hyperplane. The rigidity result also extends
when mapping to hyperquadrics in infinite dimensional Hilbert-space.
