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Given a well-ordered semi-group $\Gamma$ with a minimal system of generatorsof ordinal type at most $\omega n$ and of rational rank $r$, which satisfies apositivity and increasing condition, we construct a zero-dimensional valuationcentered on the ring of polynomials with $r$ variables such that the semi-groupof the values of the polynomial ring is equal to $\Gamma$. The constructionuses a generalization of Favre and Jonsson's version of MacLane's sequence ofkey-polynomials.

Realization of a certain class of semi-groups as value semi-groups of valuations