The Dixmier conjecture and the shape of possible counterexamples II

TL;DR

This paper advances the understanding of the Dixmier conjecture by improving bounds on irreducible pairs and extending previous results to a broader class of algebras, aiming to identify potential counterexamples.

Contribution

It improves the lower bound on the gcd of degrees of irreducible pairs and extends previous results to a family of extended algebras.

Findings

Lower bound on gcd of degrees increased to 15

Main results valid for extended algebra family

Progress towards characterizing counterexamples

Abstract

We continue with the investigation began in "The Dixmier conjecture and the shape of possible counterexamples". In that paper we introduced the notion of an irreducible pair (P,Q) as the image of the pair (X,Y) of the canonical generators of W via an endomorphism which is not an automorphism, such that it cannot be made "smaller", we let B denote the minimum of the greatest common divisor of the total degrees of P and Q, where (P,Q) runs on the irreducible pairs and we prove that $\ge $. In the present work we improve this lower bound by proving that B\ge 15. In order to do this we need to show the the main results of our previous paper remain valid for a family of algebras (W^{(l)})_{l\in \mathds{N}} that extend W.