Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields

We give a de nition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp((t)), which works uniformly for all p and all nite eld extensions of these elds, and in many other Henselian valued elds as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modi cation of the language of Macintyre. Furthermore, we show the negative result th...

On Some Dynamical Systems in Finite Fields and Residue Rings

We use character sums to confirm several recent conjectures of V. I. Arnold on the uniformity of distribution properties of a certain dynamical system in a finite field. On the other hand, we show that some conjectures are wrong. We also analyze several other conjectures of V. I. Arnold related to the orbit length of similar dynamical systems in residue rings and outline possible ways to prove ...

Definable Henselian Valuation Rings

We give model theoretic criteria for the existence of∃∀ and∀∃formulas in the ring language to define uniformly the valuation rings O of models (K,O) of an elementary theory Σ of henselian valued fields. As one of the applications we obtain the existence of an ∃∀-formula defining uniformly the valuation rings O of valued henselian fields (K,O) whose residue class field k is finite, pseudofinite,...

Commutative Rings and Finite Fields

Commutative Rings with Finite Quotient Fields

We consider the class of all commutative reduced rings for which there exists a finite subset T ⊂ A such that all projections on quotients by prime ideals of A are surjective when restricted to T . A complete structure theorem is given for this class of rings, and it is studied its relation with other finiteness conditions on the quotients of a ring over its prime ideals. Introduction Our aim i...