Reduction Theorems for the Strong Real Jacobian Conjecture

TL;DR

This paper demonstrates that certain known reductions of the Strong Real Jacobian Conjecture preserve key algebraic and geometric properties, enabling focused approaches to specific classes of polynomial maps.

Contribution

It shows that reductions to cubic homogeneous or gradient maps maintain important properties, facilitating targeted investigations of the SRJC for specific polynomial endomorphisms.

Findings

Reductions preserve algebraic properties of maps.

Reductions maintain geometric properties of maps.

Enables formulation of SRJC for specific polynomial classes.

Abstract

Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC, for instance those that induce rational function field extensions of a given fixed odd degree.