Pluckerians twisted with linear forms and Druzkowski maps

TL;DR

This paper introduces Plückerian polynomials, a new class related to twisted linear forms and Druzkowski maps, to analyze the Jacobian Conjecture through algebraic identities and linear equations.

Contribution

It defines Plückerian polynomials with increased powers and twisted forms, connecting them to Druzkowski maps and the Jacobian Conjecture for new analytical approaches.

Findings

Plückerian polynomials exhibit novel algebraic identities.

They facilitate the study of solutions to linear equations in the Jacobian condition.

Application to Druzkowski maps offers insights into the Jacobian Conjecture.

Abstract

We introduce a class of so-called Pl$\ddot{\mathrm{u}}$cker polynomials with respect to $2l\times l$ matrices, which varies the standard quadratic Pl$\ddot{\mathrm{u}}$cker expression by increased power and twisted linear forms. Besides general interests exhibited by novel algebraic identities and delicate nested structures, these polynomials fit into Dru$\dot{z}$kowski's well-known reduction of the Jacobian Conjecture. The core jacobian condition therein breaks into homogeneous linear equations with polynomial coefficients, and the Pl$\ddot{\mathrm{u}}$cker polynomials are applied to study both existence and expression of their nontrivial solutions.