Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Non-expansion in polynomial automorphisms of $\mathbb{C}^2$

Martin Bays, Tingxiang Zou

Abstract

We treat the higher-dimensional Elekes-Szabó problem in the case of the action of Aut(C^2) on C^2.

Non-expansion in polynomial automorphisms of $\mathbb{C}^2$

Abstract

We treat the higher-dimensional Elekes-Szabó problem in the case of the action of Aut(C^2) on C^2.

Paper Structure

This paper contains 6 sections, 12 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Group structure of ${\operatorname{Aut}}(k^2)$
  3. Nilpotent algebraic subgroups
  4. Correspondence triangles with bounded dimension families of infinite fixed point sets
  5. ${\operatorname{Aut}}(K^2)$
  6. Finitary consequences

Key Result

Theorem 1.2

Let $F(x,y,\bar{z}),G(x,y,\bar{z})\in{\mathbb{C}}[x,y,\bar{z}]\setminus {\mathbb{C}}[x,y]$. Suppose $(F,G)$ is not co-ordinate separable. Then for any $\varepsilon > 0$, there is $\eta>0$ such that for all finite sets $A\subseteq {\mathbb{C}}^2$ and $B\subseteq {\mathbb{C}}^{|\bar{z}|}$ with we have $|(F,G)(B)*A| := |\{(F(a,b),G(a,b)):a\in A,b\in B\}|\geq |A|^{1+\eta}$.

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • proof
  • ...and 19 more