Complexity results for CR mappings between spheres

Abstract

Using elementary number theory, we prove several results about the complexity of CR mappings between spheres. It is known that CR mappings between spheres, invariant under finite groups, lead to sharp bounds for degree estimates on real polynomials constant on a hyperplane. We show here that there are infinitely many degrees for which the uniqueness of sharp examples fails. The proof uses a Pell equation and complicated explicit computations. We also show that the so-called gap phenomenon for proper mappings between balls does not occur beyond a certain target dimension. This proof uses the solution of the postage stamp problem.

15 pages, latex, improved corollary 2

References in corpus (1)

• Regularity of CR mappings between algebraic hypersurfaces

Cited by in corpus (10)

• Normal forms, Hermitian operators, and CR maps of spheres and hyperquadrics
• Homotopy equivalence for proper holomorphic mappings
• Hermitian symmetric polynomials and CR complexity
• Bounding the rank of Hermitian forms and rigidity for CR mappings of hyperquadrics
• Uniqueness of certain polynomials constant on a line
• Rational sphere maps, linear programming, and compressed sensing
• On the Third Gap for Proper Holomorphic Maps between Balls
• Complexity of holomorphic maps from the complex unit ball to classical domains
• Polynomials constant on a hyperplane and CR maps of hyperquadrics
• On highly degenerate CR maps of spheres