Submission date: 10 October 2019

Abstract:

Comments: 19 pages, this is the new version of arxiv manuscript 1904.1715, which is now withdrawn.

Using tools from o-minimality, we prove that for two bivariate polynomials P(x,y) and Q(x,y) with coefficients in ℝ or ℂ to simultaneously exhibit small expansion, they must exploit the underlying additive or multiplicative structure of the field in nearly identical fashion. This in particular generalizes the main result of Shen and yields an Elekes--Ronyai type structural result for symmetric non-expanders, resolving an issue mentioned by de Zeeuw in [survey]. Our result also places sum-product phenomena into a more general picture of model-theoretic interest.

Mathematics Subject Classification: 03C64, 05D10

Keywords and phrases:

Full text arXiv 1910.04904: pdf, ps.