  Properties of Hermitian forms are used to investigate several natural
questions from CR Geometry. To each Hermitian symmetric polynomial we assign a
Hermitian form. We study how the signature pairs of two Hermitian forms behave
under the polynomial product. We show, except for three trivial cases, that
every signature pair can be obtained from the product of two indefinite forms.
We provide several new applications to the complexity theory of rational
mappings between hyperquadrics, including a stability result about the
existence of non-trivial rational mappings from a sphere to a hyperquadric with
a given signature pair.
