Value groups of Henselian valuations

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

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

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

Given a well-ordered semi-group Γ with a minimal system of generators of ordinal type at most ωn 1 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 semigroup of the values of the polynomial ring is equal to Γ. The construction uses a generalization of Favre ...

Strongly Dependent Ordered Abelian Groups and Henselian Fields

Strongly dependent ordered abelian groups have finite dprank. They are precisely those groups with finite spines and |{p prime : [G : pG] = ∞}| < ∞. We apply this to show that if K is a strongly dependent field, then (K, v) is strongly dependent for any henselian valuation v and the value group and residue field are stably embedded as pure structures.

Pairs of valuations and the geometry of soluble groups

We introduce the concepts of a pair of valuations and a good generating set and show how they can be used to prove geometric properties of soluble groups.