A note on complete evolution algebras

TL;DR

This paper addresses conjectures in the classification of complete evolution algebras, using algebraic geometry tools to analyze polynomial systems, and introduces new results on subalgebras, idempotents, and a conjecture on solvable evolution algebras.

Contribution

It provides positive solutions to two classification conjectures and proposes a new conjecture characterizing solvable evolution algebras.

Findings

Confirmed conjectures on classification of complete evolution algebras.

Derived new results on subalgebras and idempotents.

Proposed a conjecture for characterizing solvable evolution algebras.

Abstract

This short note provides positive answers to two conjectures of Camacho, Khudoyberdiyev, and Omirov on the classification of complete evolution algebras. Our approach is based on analysing the solution set of a generic non-linear polynomial system of equations using elementary tools from algebraic geometry. We also obtain new results on subalgebras and idempotents of evolution algebras, and conclude by proposing a conjecture that may characterise solvable evolution algebras.