Pluckerians twisted with linear forms and Druzkowski maps

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.