Realization of a Certain Class of Semi-Groups as Value Semi-Groups of Valuations

Given a well-ordered semi-group with a minimal system of generators of ordinal type at most! n1 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. The construction uses a generalization of Favre and Jonsson’s version of MacLane’s sequence of key-polynomials [3].