A new characterization of the invertibility of polynomial maps

TL;DR

This paper offers a new characterization of polynomial map invertibility, providing an equivalent condition to the Jacobian conjecture and a formula for the inverse using recursively defined polynomial sequences.

Contribution

It introduces a novel recursive polynomial sequence approach that characterizes invertibility and yields an explicit inverse formula, advancing understanding of the Jacobian conjecture.

Findings

Equivalent condition for invertibility of polynomial maps

Explicit formula for polynomial map inverses

Illustrative examples demonstrating the approach

Abstract

In this paper we present an equivalent statement to the Jacobian conjecture. For a polynomial map F on an affine space of dimension n, we define recursively n finite sequences of polynomials. We give an equivalent condition to the invertibility of F as well as a formula for the inverse of F in terms of these finite sequences of polynomials. Some examples illustrate the effective aspects of our approach.