Observations on the two dimensional Jacobian Conjecture

TL;DR

This paper proves the two-dimensional Jacobian Conjecture in specific cases where the degree of one component is at most two or the degrees of monomials share the same parity, advancing understanding of invertibility conditions.

Contribution

It establishes the invertibility of polynomial maps with invertible Jacobian under two special conditions, extending known cases of the Jacobian Conjecture.

Findings

Invertibility holds when degree of f(x) ≤ 2.

Invertibility holds when monomials in f(x) have degrees of same parity.

No restrictions on degree or parity of f(y) in these cases.

Abstract

The two dimensional Jacobian Conjecture says that a morphism $f:\mathbb{C}[x,y]\to \mathbb{C}[x,y]$ having an invertible Jacobian, is invertible. We show that a morphism $f$ having an invertible Jacobian is invertible, in each of the following two special cases: The degree of $f(x)$ is $\leq 2$; The $(0,1)$-degrees or $(1,0)$-degrees of all monomials in $f(x)$ are of the same parity. In each case there is no restriction on the degree of $f(y)$ nor on the parity of the $(0,1)$-degrees or $(1,0)$-degrees of its monomials.