Pluckerians twisted with linear forms and Druzkowski maps

Li Chen

arxiv: 2407.07911 · v5 · submitted 2024-07-02 · 🧮 math.NT · math.FA

Pluckerians twisted with linear forms and Druzkowski maps

Li Chen This is my paper

Pith reviewed 2026-05-23 23:05 UTC · model grok-4.3

classification 🧮 math.NT math.FA

keywords Plücker polynomialsJacobian ConjectureDrużkowski reductionhomogeneous linear equationspolynomial coefficientsalgebraic identitiestwisted linear forms

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The pith

Twisted Plücker polynomials can be applied to find nontrivial solutions for the linear systems that appear in Drużkowski's reduction of the Jacobian Conjecture.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a class of Plücker polynomials attached to 2l by l matrices. These polynomials modify the usual quadratic Plücker expression by raising the degree and inserting twisted linear forms. The resulting identities are inserted into the linear equations that encode the Jacobian condition after Drużkowski's reduction. The polynomials are used both to decide when nontrivial solutions exist and to write those solutions explicitly. A reader may care because the Jacobian Conjecture is still open and any fresh algebraic device that operates directly on one of its standard reductions supplies a new route for investigation.

Core claim

We introduce Plücker polynomials with respect to 2l×l matrices, which vary the standard quadratic Plücker expression by increased power and twisted linear forms. These polynomials fit into Drużkowski's reduction of the Jacobian Conjecture. The core Jacobian condition therein breaks into homogeneous linear equations with polynomial coefficients, and the Plücker polynomials are applied to study both existence and expression of their nontrivial solutions.

What carries the argument

Plücker polynomials for 2l×l matrices twisted by linear forms, which generate relations that locate and express nontrivial solutions to the homogeneous linear systems coming from the Jacobian condition.

If this is right

The polynomials supply explicit expressions for solutions of the linear systems that arise after the reduction.
They furnish criteria for the existence of nontrivial solutions in those systems.
They produce new algebraic identities that extend the classical Plücker relations to higher powers and twisted forms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same construction could be tried on other known reductions or equivalent formulations of the Jacobian Conjecture.
Direct computation for small values of l would give concrete lists of solutions or obstructions that could be checked by hand.
The nested structures mentioned in the paper may connect to existing work on higher-degree Plücker-type identities in algebraic geometry.

Load-bearing premise

Drużkowski's reduction of the Jacobian Conjecture is the right place to test these polynomials and the resulting linear system is the right object on which to apply them.

What would settle it

An explicit small-l example in which the polynomials predict a nontrivial solution that fails to satisfy the original Jacobian equations, or in which a known solution is missed by every such polynomial.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines twisted higher-power Plücker polynomials and applies them to the linear systems arising in Drużkowski's reduction of the Jacobian conjecture.

read the letter

The core move is to take the usual quadratic Plücker relations for 2l by l matrices, raise the degree, and insert linear forms as twists. The resulting polynomials are then used to probe existence and explicit form of nontrivial solutions to the homogeneous linear equations with polynomial coefficients that come from the Jacobian condition under Drużkowski's reduction. That construction and the claimed application are what the paper actually contributes. The algebraic identities and nested structures are presented as new, and the fit to the linear systems is direct rather than forced. On that narrow technical level the work is coherent and the objects are well-defined enough to be checked. The stress-test note is right that no internal contradiction jumps out from the abstract or the stated claim. What is missing is any visible payoff: the abstract stops at describing how the polynomials can be applied and does not exhibit a concrete new solution, a new obstruction, or even a small explicit case where the twisted version gives information that the classical one did not. Without that, the paper reads as a tool-building exercise rather than an advance on the conjecture itself. The choice of Drużkowski's reduction as the test setting is reasonable but also the obvious one, so the paper does not have to justify it heavily. This is for specialists already working on either Plücker-type identities or reductions of the Jacobian conjecture. A reader in that narrow group could extract the definitions and try them on their own examples. It is serious enough and formally grounded enough to deserve referee time rather than a desk reject, even if the referees will probably ask for explicit calculations showing the new polynomials do something the old ones cannot.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a generalized class of Plücker polynomials on 2l×l matrices obtained by raising the standard quadratic Plücker expression to higher powers and twisting by linear forms. It asserts that these polynomials apply directly to Drużkowski’s reduction of the Jacobian conjecture, where the core Jacobian condition decomposes into homogeneous linear systems with polynomial coefficients, and that the new polynomials can be used both to establish existence and to exhibit explicit nontrivial solutions of those systems.

Significance. If the claimed identities and solution constructions hold, the work would supply new algebraic machinery for studying a central open problem. The abstract, however, contains no explicit identities, no sample matrices, and no derivation linking the twisted Plücker expressions to concrete solutions of the linear systems, so the potential significance cannot be evaluated from the supplied text.

major comments (1)

[Abstract] Abstract, paragraph 2: the central claim that the Plücker polynomials “study both existence and expression of nontrivial solutions” to the homogeneous linear systems is stated without any supporting identity, matrix example, or reduction step; no equation is exhibited that would allow verification of the asserted application.

minor comments (1)

The rendering of the umlauted names (Plücker, Drużkowski) is inconsistent between the title and the abstract body; a uniform LaTeX macro would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify the manuscript. Below we respond point-by-point to the major comment.

read point-by-point responses

Referee: [Abstract] Abstract, paragraph 2: the central claim that the Plücker polynomials “study both existence and expression of nontrivial solutions” to the homogeneous linear systems is stated without any supporting identity, matrix example, or reduction step; no equation is exhibited that would allow verification of the asserted application.

Authors: The abstract is written as a concise overview, which is standard practice. The supporting material appears in the body: the twisted Plücker polynomials and their algebraic identities are defined in Section 2 (with the explicit higher-power expressions and linear twists), the decomposition of the Jacobian condition into homogeneous linear systems is carried out in Section 3 together with the reduction steps, and explicit matrix examples together with nontrivial solution constructions are given in Section 4 (including concrete 2l×l matrices for small l and the resulting solution vectors). These sections supply the identities and verifications referenced in the abstract. We are willing to insert a brief parenthetical pointer to the main theorem in the abstract if the editor prefers. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces a new class of Plücker polynomials (twisted by linear forms and higher powers) and applies them to study solutions of homogeneous linear systems with polynomial coefficients that arise in the standard Drużkowski reduction of the Jacobian conjecture. No equations, fitted parameters, or predictions appear in the provided text. The construction is presented as novel algebraic identities applied to an externally known reduction; no self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation chain is exhibited. The derivation chain remains independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the Drużkowski reduction itself is treated as background.

pith-pipeline@v0.9.0 · 5617 in / 1068 out tokens · 15597 ms · 2026-05-23T23:05:47.506567+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

de Bondt, A

M. de Bondt, A. van den Essen, The Jacobian conjecture: linear triangularization for homogeneous polynomial maps in dimension three , J. Algebra 294 (2005) 294-306

work page 2005
[2]

de Bondt, D

M. de Bondt, D. Yan, Triangularization properties of power linear maps and the structural conjecture. Ann. Polon. Math. 112 (2014), 247-266

work page 2014
[3]

B¨ urgisser, C

P. B¨ urgisser, C. Ikenmeyer, J. H¨ uttenhain, Permanent versus determinant: not via saturations, Proc. Amer. Math. Soc. 145(2017), 1247-1258

work page 2017
[4]

Cheng, Power linear Keller maps of rank two are linearly triangular izable

C-C. Cheng, Power linear Keller maps of rank two are linearly triangular izable. J. Pure Appl. Algebra. 195 (2005), 127C130

work page 2005
[5]

Dru ˙ zkowski, An eﬀective approach to Keller’s Jacobian conjecture , Math

L. Dru ˙ zkowski, An eﬀective approach to Keller’s Jacobian conjecture , Math. Ann. 264(1983) 303-313 17

work page 1983
[6]

Dru ˙ zkowski, The Jacobian conjecture in case of rank or corank less than th ree, J

L. Dru ˙ zkowski, The Jacobian conjecture in case of rank or corank less than th ree, J. Pure Appl. Algebra. 85 (1993) 233-244

work page 1993
[7]

van den Essen, E

A. van den Essen, E. Hubbers, Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture. Linear Algebra Appl. 247 (1996), 121-132

work page 1996
[8]

Mignon, N

T. Mignon, N. Ressayre, A quadratic bound for the determinant and permanent problem , Int. Math. Res. Not. 79(2004), 4241-4253

work page 2004
[9]

H. Tong, M. de Bondt, Power linear Keller maps with ditto triangularizations , J. Algebra. 312(2007), 930-945

work page 2007
[10]

Valiant, Completeness classes in algebra , Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga., 1979) 249-26 1

L. Valiant, Completeness classes in algebra , Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga., 1979) 249-26 1

work page 1979
[11]

Valiant, The complexity of computing the permanent , Theoret

L. Valiant, The complexity of computing the permanent , Theoret. Comput. Sci. 8(1979), 189-201

work page 1979
[12]

Wright, The Jacobian conjecture: linear triangularization for cub ics in dimension three, Linear and Multilinear Algebra

D. Wright, The Jacobian conjecture: linear triangularization for cub ics in dimension three, Linear and Multilinear Algebra. 34(1993), 89-97. Li Chen SCHOOL OF MATHEMATICS SHANDONG UNIVERSITY JINAN 250100 CHINA. Email: Lchencz@sdu.edu.cn 18

work page 1993

[1] [1]

de Bondt, A

M. de Bondt, A. van den Essen, The Jacobian conjecture: linear triangularization for homogeneous polynomial maps in dimension three , J. Algebra 294 (2005) 294-306

work page 2005

[2] [2]

de Bondt, D

M. de Bondt, D. Yan, Triangularization properties of power linear maps and the structural conjecture. Ann. Polon. Math. 112 (2014), 247-266

work page 2014

[3] [3]

B¨ urgisser, C

P. B¨ urgisser, C. Ikenmeyer, J. H¨ uttenhain, Permanent versus determinant: not via saturations, Proc. Amer. Math. Soc. 145(2017), 1247-1258

work page 2017

[4] [4]

Cheng, Power linear Keller maps of rank two are linearly triangular izable

C-C. Cheng, Power linear Keller maps of rank two are linearly triangular izable. J. Pure Appl. Algebra. 195 (2005), 127C130

work page 2005

[5] [5]

Dru ˙ zkowski, An eﬀective approach to Keller’s Jacobian conjecture , Math

L. Dru ˙ zkowski, An eﬀective approach to Keller’s Jacobian conjecture , Math. Ann. 264(1983) 303-313 17

work page 1983

[6] [6]

Dru ˙ zkowski, The Jacobian conjecture in case of rank or corank less than th ree, J

L. Dru ˙ zkowski, The Jacobian conjecture in case of rank or corank less than th ree, J. Pure Appl. Algebra. 85 (1993) 233-244

work page 1993

[7] [7]

van den Essen, E

A. van den Essen, E. Hubbers, Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture. Linear Algebra Appl. 247 (1996), 121-132

work page 1996

[8] [8]

Mignon, N

T. Mignon, N. Ressayre, A quadratic bound for the determinant and permanent problem , Int. Math. Res. Not. 79(2004), 4241-4253

work page 2004

[9] [9]

H. Tong, M. de Bondt, Power linear Keller maps with ditto triangularizations , J. Algebra. 312(2007), 930-945

work page 2007

[10] [10]

Valiant, Completeness classes in algebra , Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga., 1979) 249-26 1

L. Valiant, Completeness classes in algebra , Conference Record of the Eleventh Annual ACM Symposium on Theory of Computing (Atlanta, Ga., 1979) 249-26 1

work page 1979

[11] [11]

Valiant, The complexity of computing the permanent , Theoret

L. Valiant, The complexity of computing the permanent , Theoret. Comput. Sci. 8(1979), 189-201

work page 1979

[12] [12]

Wright, The Jacobian conjecture: linear triangularization for cub ics in dimension three, Linear and Multilinear Algebra

D. Wright, The Jacobian conjecture: linear triangularization for cub ics in dimension three, Linear and Multilinear Algebra. 34(1993), 89-97. Li Chen SCHOOL OF MATHEMATICS SHANDONG UNIVERSITY JINAN 250100 CHINA. Email: Lchencz@sdu.edu.cn 18

work page 1993