An effective approach to Picard-Vessiot theory and the Jacobian Conjecture

TL;DR

This paper links the Jacobian Conjecture to Picard-Vessiot extensions, providing a simplified criterion and bounds for verifying polynomial automorphisms, especially effective for cubic homogeneous mappings.

Contribution

It offers a new simplified criterion and bounds for the Jacobian Conjecture using Picard-Vessiot theory, improving detection of polynomial automorphisms.

Findings

Established an equivalent statement of the Jacobian Conjecture via Picard-Vessiot extensions.

Provided a simplified criterion for polynomial automorphisms with a non-zero constant Jacobian.

Derived bounds on the number of Wronskian determinants needed for verification.

Abstract

In this paper we present a theorem concerning an equivalent statement of the Jacobian Conjecture in terms of Picard-Vessiot extensions. Our theorem completes the earlier work of T. Crespo and Z. Hajto which suggested an effective criterion for detecting polynomial automorphisms of affine spaces. We show a simplified criterion and give a bound on the number of wronskians determinants which we need to consider in order to check if a given polynomial mapping with non-zero constant Jacobian determinant is a polynomial automorphism. Our method is specially efficient with cubic homogeneous mappings introduced and studied in fundamental papers by H. Bass, E. Connell, D. Wright and L. Druzkowski.