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

TL;DR

This paper constructs zero-dimensional valuations with prescribed semi-groups of values, extending MacLane's key-polynomial sequences to a broader class of semi-groups with specific order and rank properties.

Contribution

It generalizes Favre and Jonsson's approach to realize a class of semi-groups as valuation value semi-groups, broadening the scope of valuation construction methods.

Findings

Successfully constructs valuations for semi-groups with bounded ordinal type and rational rank.

Extends MacLane's key-polynomial sequence method to new semi-group classes.

Provides a framework for realizing semi-groups as valuation value semi-groups.

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.