The Dixmier conjecture and the shape of possible counterexamples

TL;DR

This paper establishes a lower bound on the size of potential counterexamples to the Dixmier Conjecture, specifically showing that the minimum gcd of degrees must exceed 15.

Contribution

It provides a new lower bound on the degrees of possible counterexamples to the Dixmier Conjecture, narrowing the search for such counterexamples.

Findings

Minimum gcd of degrees in counterexamples exceeds 15

Provides constraints on the shape of potential counterexamples

Advances understanding of the structure of possible counterexamples

Abstract

We establish a lower bound for the size of possible counterexamples of the Dixmier Conjecture. We prove that $B>15$, where $B$ is the minimum of the greatest common divisor of the total degrees of $P$ and $Q$, where $(P,Q)$ runs over the counterexamples of the Dixmier Conjecture.