Some uniform effective results on Andr\'{e}--Oort for sums of powers in Cⁿ

We prove an Andr\'e--Oort-type result for a family of hypersurfaces in Cⁿ that is both uniform and effective. Let K_* denote the single exceptional imaginary quadratic field which occurs in the Siegel--Tatuzawa lower bound for the class number. We prove that, for m, n ∈ Z>0, there exists an effective constant $c(m, n)>0$ with the following property: if pairwise distinct singular moduli x₁, …, xₙ with respective discriminants Δ₁, …, Δₙ are such that a₁ x₁ᵐ + … + aₙ xₙᵐ ∈ Q for some a₁, …, aₙ ∈ Q \0\ and # \ Δᵢ : Q(√Δᵢ) = K_*\ ≤ 1, then maxᵢ Δᵢ ≤ c(m, n). In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples (x₁, x₂, x₃) of singular moduli such that a₁ x₁ + a₂ x₂ + a₃ x₃ ∈ Q for some a₁, a₂, a₃ ∈ Q \0\.