Eliminating field quantifiers in strongly dependent henselian fields

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.

Voting by Eliminating Quantifiers

Mathematical theory of voting and social choice has attracted much attention. In the general setting one can view social choice as a method of aggregating individual, often conflicting preferences and making a choice that is the best compromise. How preferences are expressed and what is the “best compromise” varies and heavily depends on a particular situation. The method we propose in this pap...

Henselian Function Fields

HENSELIAN RESIDUALLY p-ADICALLY CLOSED FIELDS

In (Arch. Math. 57 (1991), pp. 446–455), R. Farré proved a positivstellensatz for real-series closed fields. Here we consider p-valued fields 〈K, vp〉 with a non-trivial valuation v which satisfies a compatibility condition between vp and v. We use this notion to establish the p-adic analogue of real-series closed fields; these fields are called henselian residually p-adically closed fields. Fir...

Automorphisms and Isomorphisms of Real Henselian Fields

Let K and L be ordered algebraic extensions of an ordered field F. Suppose K and L are Henselian with Archimedean real closed residue class fields. Then K and L are shown to be F-isomorphic as ordered fields if they have the same value group. Analogues to this result are proved involving orderings of higher level, unordered extensions, and, when K and L are maximal valued fields, transcendental...