Lectures on Geometry and Topology of Polynomials - Surrounding the Jacobian Conjecture

TL;DR

This paper discusses the geometric and topological properties of polynomials in two variables within affine algebraic geometry, focusing on the longstanding Jacobian Conjecture and recent theoretical developments.

Contribution

It provides an overview of recent advances in affine algebraic geometry related to the Jacobian Conjecture and explores geometric and topological aspects of polynomials in two variables.

Findings

Insights into the geometric structure of polynomials in two variables.

Discussion of the unresolved Jacobian Conjecture since 1939.

Overview of recent theoretical progress in affine algebraic geometry.

Abstract

Recent developments of affine algebraic geometry, especially the theory of open algebraic surfaces, provide means to systematically explore geometric and topological properties of polynomials in two variables. Nevertheless, there is one unsurmountable problem remained even in the case of two variables, which has been unsolved since 1939, that is the Jacobian Conjecture. These are notes for author's lectures on the geometry and topology of polynomials and the Jacobian Conjecture delivered at the l'Universit\'e de Bordeaux I and at Osaka University in 2003.