Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Number of homogeneous components of counterexamples to the Dixmier conjecture

Jorge Guccione, Juan Jose Guccione, Christian Valqui

Abstract

Assume that $P$ and $Q$ are elements of $A_1$ satisfying $[P,Q] = 1$. The Dixmier Conjecture for $A_1$ says that they always generate $A_1$. We show that if $P$ is a sum of not more than~$4$ homogeneous elements of $A_1$ then $P$ and $Q$ generate $A_1$, which generalizes the main result in arXiv:2210.00257.

Number of homogeneous components of counterexamples to the Dixmier conjecture

Abstract

Assume that and are elements of satisfying . The Dixmier Conjecture for says that they always generate . We show that if is a sum of not more than~ homogeneous elements of then and generate , which generalizes the main result in arXiv:2210.00257.
Paper Structure (4 sections, 9 theorems, 82 equations)

This paper contains 4 sections, 9 theorems, 82 equations.

Table of Contents

  1. Preliminaries
  2. Support of univariate polynomials
  3. Cases
  4. Lower bound for $m(P)$

Key Result

Proposition 1.2

Assume that $(P,Q)$ is a counterexample to the DC (this means that $P$ and $Q$ do not generate $A_1$ and that $[P,Q] = 1$). Then, we have $v_{1,-1}(P)>0$ and $v_{-1,1}(P)>0$.

Theorems & Definitions (25)

  • Remark 1.1
  • Proposition 1.2
  • proof
  • Remark 1.3
  • Proposition 1.4
  • proof
  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 15 more