Symmetric Jacobians

TL;DR

This paper investigates polynomial maps with symmetric or antisymmetric Jacobians, exploring their relation to the Jacobian Conjecture and whether focusing on such maps suffices for a proof.

Contribution

It demonstrates that proving the Jacobian Conjecture for polynomial maps with symmetric Jacobians of a specific form is sufficient to establish the conjecture in general.

Findings

Proves sufficiency of verifying the Jacobian Conjecture for symmetric Jacobian maps.

Shows that maps with symmetric Jacobians of the form JH are central to the conjecture.

Establishes conditions under which the Jacobian Conjecture can be reduced to symmetric Jacobian cases.

Abstract

This article is about polynomial maps with a certain symmetry and/or antisymmetry in their Jacobians, and whether the Jacobian Conjecture is satisfied for such maps, or whether it is sufficient to prove the Jacobian Conjecture for such maps. For instance, we show that it suffices to prove the Jacobian conjecture for polynomial maps x + H over C such that JH satisfies all symmetries of the square, where H is homogeneous of arbitrary degree d >= 3.