An asymmetric version of Elekes-Szabó via group actions

Abstract: Elekes and Rónyai showed that a bivariate real polynomial f(x,y) expands, meaning $|f(A,A)| \geq c|A|^{1+\eta}$ for all finite A, unless f can be written as the composition of addition or multiplication with univariate polynomials. The proof can be seen as going via the group configuration theorem of model theory. I will talk about recent work with Tingxiang Zou, in which we consider a more general setup where we can apply a homogeneous space version of this group configuration theorem, and yet still subsequently reduce to an abelian group. We deduce asymmetric expansion $|f(A,B)| \geq c|A|^{1+\eta}$ even when B is allowed to be drastically smaller than A. Moreover, we obtain a similar result when y is allowed to be a tuple. Allowing x to also be a tuple introduces new phenomena, and if time permits I may mention some partial results in this case.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.