# Frameworks for two-dimensional Keller maps

## TL;DR

This paper discusses frameworks for analyzing two-dimensional Keller maps, which are potential counterexamples to the Jacobian Conjecture, by translating the problem into combinatorial conditions on Picard groups.

## Contribution

It introduces and discusses multiple frameworks for understanding Keller maps in dimension two, expanding on previous work with more complex examples.

## Key findings

- Several frameworks for Keller maps are described.
- New, more complex frameworks are introduced.
- The frameworks facilitate combinatorial analysis of Keller maps.

## Abstract

A Keller map is a counterexample to the Jacobian Conjecture. In dimension two every such map, if exists, leads to a complicated set of conditions on the map between the Picard groups of suitable compactifications of the affine plane. This is essentially a combinatorial problem. Some solutions to it, that we call frameworks, are described and discussed. This second version of the paper includes several more frameworks than the first version, including one substantially more complicated framework.