Polynomials constant on a hyperplane and CR maps of hyperquadrics

We prove a sharp degree bound for polynomials constant on a hyperplane with a fixed number of distinct monomials for dimensions 2 and 3. We study the connection with monomial CR maps of hyperquadrics and prove similar bounds in this setup with emphasis on the case of spheres. The results support generalizing a conjecture on the degree bounds to the more general case of hyperquadrics. Keywords. Polynomials constant on a hyperplane, CR mappings of spheres and hyperquadrics, monomial mappings, degree estimates, Newton diagram. 2000 Mathematics Subject Classification. 14P99, 05A20, 32H35, 11C08