Ideas about the Jacobian Conjecture

TL;DR

This paper explores new ideas related to the Jacobian Conjecture, proposing a degree bound conjecture for field extensions and suggesting alternative assumptions to known theorems to advance understanding of the problem.

Contribution

It introduces a conjecture linking the degree of field extensions to polynomial degrees and proposes replacing degree assumptions with minimal polynomial degree conditions in existing theorems.

Findings

Proposes a conjecture that if true, implies the generalized Jacobian Conjecture.

Suggests replacing degree bounds with minimal polynomial degree assumptions in known results.

Provides analogous results under new assumptions.

Abstract

Let $F:\mathbb{C}[x_1,\ldots,x_n] \to \mathbb{C}[x_1,\ldots,x_n]$ be a $\mathbb{C}$-algebra endomorphism that has an invertible Jacobian. We bring two ideas concerning the Jacobian Conjecture: First, we conjecture that for all $n$, the degree of the field extension $\mathbb{C}(F(x_1),\ldots,F(x_n)) \subseteq \mathbb{C}(x_1,\ldots,x_n)$ is less than or equal to $d^{n-1}$, where $d$ is the minimum of the degrees of the $F(x_i)$'s. If this conjecture is true, then the generalized Jacobian Conjecture is true. Second, we suggest to replace in some known theorems the assumption on the degrees of the $F(x_i)$'s by a similar assumption on the degrees of the minimal polynomials of the $x_i$'s over $\mathbb{C}(F(x_1)\ldots,f(x_n))$; this way we obtain some analogous results to the known ones.