Generalization of the ABC theorem on locally nilpotent derivations

TL;DR

This paper generalizes the ABC Theorem for locally nilpotent derivations to specific polynomial forms and explores applications in constructing rigid algebras and analyzing their invariants.

Contribution

It extends the ABC Theorem to polynomials with monomials involving distinct variables, enabling new algebraic constructions and invariant descriptions.

Findings

Generalized the ABC Theorem for a new class of polynomials

Constructed examples of rigid and semi-rigid algebras

Described the Makar-Limanov invariant for specific algebra forms

Abstract

We obtain a generalization of the ABC Theorem on locally nilpotent derivations to the case of the polynomials with m monomials such that each variable is included just in one monomial. As applications of this result we provide some construction of rigid and semi-rigid algebras and describe the Makar-Limanov invariant of algebras of a special form.