Characterizing solvable evolution algebras via absence of idempotents and triviality of the idempotent system

Prove that for any complex evolution algebra E, the following conditions are equivalent: (i) E is solvable; (ii) E admits no idempotent element; and (iii) the system M_B(E)^t [x_1^2, …, x_n^2]^t = [x_1, …, x_n]^t admits only the trivial solution.

Background

The authors propose a new conjecture aiming to characterize solvable evolution algebras. They note that the absence of idempotents is necessary for solvability and provide supporting evidence in dimensions 1 and 2.

The conjecture connects solvability to two equivalent formulations: the absence of idempotents and the uniqueness (triviality) of solutions to the system that encodes the idempotent condition via the structure matrix.

References

\begin{conjecture}\label{conj} Let $E$ be a complex evolution algebra. Then, the following assertions are equivalent: \begin{enumerate}[\rm (i)] \item $E$ is solvable; \item $E$ admits no idempotents; and \item the system~sist_3 only admits the trivial solution. \end{enumerate} \end{conjecture}

sist_3:

MB(E)t(x12⋮xn2)=(x1⋮xn).M_B(\mathcal{E})^t\begin{pmatrix} x_1^2 \\ \vdots \\ x_n^2 \end{pmatrix}=\begin{pmatrix} x_1 \\ \vdots \\ x_n \end{pmatrix}.

— A note on complete evolution algebras (2512.12418 - García-Martínez et al., 13 Dec 2025) in Conjecture, Section 3.3 (Idempotents)