Full nonzero-coordinate solution to the nonlinear system x_i^2 = Σ_j a_{ij} x_j

Prove that for any complex invertible matrix A = (a_{ij}) ∈ C^{n×n}, the system of equations x_i^2 = ∑_{j=1}^n a_{ij} x_j for i = 1, …, n admits a solution (x_1, …, x_n) with x_i ≠ 0 for all i.

Background

The paper studies a nonlinear polynomial system associated with evolution algebras. Camacho, Khudoyberdiyev, and Omirov formulated a conjecture asserting the existence of a solution where all coordinates are nonzero for any complex invertible matrix A.

This note proves a relaxed version guaranteeing a nontrivial solution with at least two nonzero coordinates, sufficient for their applications to evolution algebras, but does not resolve the original full nonzero-coordinate conjecture.

References

In its original form, the conjecture says that given a complex invertible matrix A=(a_{ij})_{1\leq i,j\leq n}, the system of equations admits a solution (x_1,x_2,\dots,x_n) such that x_i\neq0 for all i.

— A note on complete evolution algebras (2512.12418 - García-Martínez et al., 13 Dec 2025) in Section 2 (The key result)