Complexity results for CR mappings between spheres

Details Complexity results for CR mappings between spheres Complexity results for CR mappings between spheres

2007-08-23

arxiv.org/abs/0708.3232v2

International Journal of Mathematics, 20 (2009), no. 2, 149-166

Mathematics

Complex Variables

15 pages, latex, improved corollary 2

Scientific paper

10.1142/S0129167X09005248

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.

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