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

Abstract

Given a well-ordered semi-group $\Gamma$ with a minimal system of generators of ordinal type at most $\omega n$ 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 $\Gamma$. The construction uses a generalization of Favre and Jonsson's version of MacLane's sequence of key-polynomials.