Abstract. We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of SU (2). We introduce the asymptotic positivity ratio and compute it for cyclic subgroups of U (2). We calculate the signature pair for dihedral subgroups of U (2).

Abstract. 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.

Several questions in CR geometry lead naturally to the study of bihomogeneous polynomials r(z,z) on C n ×C n for which r(z,z) z 2d = h(z) 2 for some natural number d and a holomorphic polynomial mapping h = (h 1 , . . . , h K ) from C n to C K . When r has this property for some d, one seeks relationships between d, K, and the signature and rank of the coefficient matrix of r. In this paper, we reformulate this basic question as a question about the growth of the Hilbert function of a homogeneous ideal in C[z 1 , . . . , zn] and apply a well-known result of Macaulay to estimate some natural quantities.

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