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A variant of Laplace expansion and linear algebra of Druzkowski map

Li Chen

TL;DR

The paper introduces a Laplace-expansion–style identity for a class of linear equations arising from Druzkowski maps linked to the Jacobian Conjecture, enabling Cramer-type solutions built from matrix minors. In the canonical $(3,3,3)$ setting, it proves an equivalence between the existence of nontrivial linear dependencies among triple products of the $f_i$ and the vanishing of a determinant constructed from the coefficients, via Theorem 20 and Theorem unee. The framework then extends to degenerate $(2,2,2)$ and upgraded $(4,4,4)$ cases, revealing an inductive factorization structure that suggests a route toward a uniform theory for these polynomial linear systems. Together, these results provide a structured, algebraic approach to solving Druzkowski-related equations and illuminate connections between determinant identities and polynomial factorization in the study of the Jacobian Conjecture.

Abstract

Druzkowski s reduction of the Jacobian Conjecture gives rise to a class of highly complicated linear equations with polynomial coefficients. We find an unrecorded algebraic identity and apply it as a variant of the Laplace determinant expansion formula to solve such equation in a way analogous to(and no less elegant than) Cramer rule. Three consecutive cases are addressed where the situation upgrades from one case to the next.

A variant of Laplace expansion and linear algebra of Druzkowski map

TL;DR

The paper introduces a Laplace-expansion–style identity for a class of linear equations arising from Druzkowski maps linked to the Jacobian Conjecture, enabling Cramer-type solutions built from matrix minors. In the canonical setting, it proves an equivalence between the existence of nontrivial linear dependencies among triple products of the and the vanishing of a determinant constructed from the coefficients, via Theorem 20 and Theorem unee. The framework then extends to degenerate and upgraded cases, revealing an inductive factorization structure that suggests a route toward a uniform theory for these polynomial linear systems. Together, these results provide a structured, algebraic approach to solving Druzkowski-related equations and illuminate connections between determinant identities and polynomial factorization in the study of the Jacobian Conjecture.

Abstract

Druzkowski s reduction of the Jacobian Conjecture gives rise to a class of highly complicated linear equations with polynomial coefficients. We find an unrecorded algebraic identity and apply it as a variant of the Laplace determinant expansion formula to solve such equation in a way analogous to(and no less elegant than) Cramer rule. Three consecutive cases are addressed where the situation upgrades from one case to the next.
Paper Structure (10 sections, 9 theorems, 54 equations)

This paper contains 10 sections, 9 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Linear equations in Dru$\dot{z}$kowski map
  3. Cramer-type solution via Laplace expansion
  4. Results and organization of this paper
  5. Standard case
  6. Degenerate and upgraded cases
  7. General problems
  8. Degenerate case
  9. Upgraded case
  10. Appendix: Proof of Theorem \ref{['unee']}($(i)\Rightarrow(ii)$)

Key Result

Theorem 1.2

The equation $A\textbf{x}=\textbf{0}$ admits nontrivial solutions if and only if $\textbf{x}=(b_2c_3-b_3c_2,a_3c_2-a_2c_3, a_2b_3-a_3b_2)^T$ is a solution(not necessarily nontrivial).

Theorems & Definitions (17)

  • Conjecture 1.1
  • Theorem 1.2
  • proof
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 3.1
  • ...and 7 more