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Properties of Hermitian forms are used to investigate several naturalquestions from CR Geometry. To each Hermitian symmetric polynomial we assign aHermitian form. We study how the signature pairs of two Hermitian forms behaveunder the polynomial product. We show, except for three trivial cases, thatevery signature pair can be obtained from the product of two indefinite forms.We provide several new applications to the complexity theory of rationalmappings between hyperquadrics, including a stability result about theexistence of non-trivial rational mappings from a sphere to a hyperquadric witha given signature pair.

Hermitian symmetric polynomials and CR complexity