Polynomial atlases on manifolds

TL;DR

This paper explores manifolds with polynomial transition maps to provide an equivalent formulation of the Jacobian conjecture over the real numbers, connecting geometric structures with a major open problem in algebraic geometry.

Contribution

It introduces polynomial atlases on manifolds as a new framework to restate the Jacobian conjecture, bridging differential geometry and algebraic geometry.

Findings

Equivalent formulation of the Jacobian conjecture using polynomial atlases

Establishment of a geometric perspective on a longstanding algebraic problem

Potential new approaches to solving the Jacobian conjecture

Abstract

We consider manifolds whose transition maps are restrictions of polynomial mappings $\mathbb{R}^n\to\mathbb{R}^n$, and use them to give an equivalent statement of the Jacobian conjecture over the real field.