Jacobian Conjecture and Nilpotency

TL;DR

This paper proves invertibility of certain polynomial endomorphisms over fields of characteristic zero, where the map is close to the identity and the Jacobian matrix's cube is zero, advancing understanding of the Jacobian Conjecture.

Contribution

It establishes invertibility for polynomial maps with specific nilpotent Jacobian conditions, using a novel inversion algorithm and proposing a related conjecture.

Findings

Proved invertibility of polynomial maps with nilpotent Jacobian cube.

Connected nilpotency degree to the steps in the inversion algorithm.

Formulated a conjecture linking Jacobian nilpotency to inversion complexity.

Abstract

For K a field of characteristic 0 and d any integer number greater than or equal to 2, we prove the invertibility of polynomial endomorphisms of the affine space of dimension d over K of the form F=Id+H, where each coordinate of H is the cube of a linear form and the cube of the Jacobian matrix of H is equal to zero. Our proof uses the inversion algorithm for polynomial maps presented in our previous paper. Our current result leads us to formulate a conjecture relating the nilpotency degree of the Jacobian matrix of H with the number of necessary steps in the inversion algorithm.