2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/161704Given a well-ordered semi-group $Γ$ with a minimal system of generators of ordinal type at most $ωn$ and of rational rank $r$, which satisfies a positivity and increasing condition, we construct a zero-dimensional valuation centered on the ring of polynomials with $r$ variables such that the semi-group of the values of the polynomial ring is equal to $Γ$. The construction uses a generalization of Favre and Jonsson's version of MacLane's sequence of key-polynomials.To be Published in BIMSAlgebraic GeometryCommutative Algebratext