Number-theoretic properties of certain CR mappings

We first prove a uniqueness result for certain group-invariant CR mappings to hyperquadrics. For cyclic groups these mappings lead to a collection of polynomials f _p,q (with integer coefficients) in two variables; here p and q are positive integers. We use the uniqueness result to prove some surprising number-theoretic results about the f _p,q, in particular, f _p,q is congruent to x ^P+y ^P modulo (p) (for P ≥ 2) if and only if p is prime. We also determine recurrence relations for these polynomials for q ≤ 3 and determine a functional equation for a generating function. Finally, we discuss the invariant polynomials that arise for certain representations of dihedral groups to illustrate the non-Abelian case.