For an algebraic number field 𝕂 and a subset ${α_1, ..., α_r} ⊆ 𝓞_{𝕂}$, we establish a lower bound for the average of the logarithmic heights that depends on the ideal of polynomials in $ℚ[x_1, ..., x_r]$ vanishing at the point $(α_1, ..., α_r)$.
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