1,080 research outputs found
An existential 0-definition of F_q[[t]] in F_q((t))
We show that the valuation ring F_q[[t]] in the local field F_q((t)) is
existentially definable in the language of rings with no parameters. The method
is to use the definition of the henselian topology following the work of
Prestel-Ziegler to give an existential-F_q-definable bounded neighbouhood of 0.
Then we `tweak' this set by subtracting, taking roots, and applying Hensel's
Lemma in order to find an existential-F_q-definable subset of F_q[[t]] which
contains tF_q[[t]]. Finally, we use the fact that F_q is defined by the formula
x^q-x=0 to extend the definition to the whole of F_q[[t]] and to rid the
definition of parameters.
Several extensions of the theorem are obtained, notably an existential
0-definition of the valuation ring of a non-trivial valuation with divisible
value group.Comment: 9 page
Some Definability Results in Abstract Kummer Theory
Let be a semiabelian variety over an algebraically closed field, and let
be an irreducible subvariety not contained in a coset of a proper algebraic
subgroup of . We show that the number of irreducible components of
is bounded uniformly in , and moreover that the bound is
uniform in families .
We prove this by purely Galois-theoretic methods. This proof applies in the
more general context of divisible abelian groups of finite Morley rank. In this
latter context, we deduce a definability result under the assumption of the
Definable Multiplicity Property (DMP). We give sufficient conditions for finite
Morley rank groups to have the DMP, and hence give examples where our
definability result holds.Comment: 21 pages; minor notational fixe
Generic Automorphisms and Green Fields
We show that the generic automorphism is axiomatisable in the green field of
Poizat (once Morleyised) as well as in the bad fields which are obtained by
collapsing this green field to finite Morley rank. As a corollary, we obtain
"bad pseudofinite fields" in characteristic 0. In both cases, we give geometric
axioms. In fact, a general framework is presented allowing this kind of
axiomatisation. We deduce from various constructibility results for algebraic
varieties in characteristic 0 that the green and bad fields fall into this
framework. Finally, we give similar results for other theories obtained by
Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories
having the definable multiplicity property. We also close a gap in the
construction of the bad field, showing that the codes may be chosen to be
families of strongly minimal sets.Comment: Some minor changes; new: a result of the paper (Cor 4.8) closes a gap
in the construction of the bad fiel
External definability and groups in NIP theories
We prove that many properties and invariants of definable groups in NIP
theories, such as definable amenability, G/G^{00}, etc., are preserved when
passing to the theory of the Shelah expansion by externally definable sets,
M^{ext}, of a model M. In the light of these results we continue the study of
the "definable topological dynamics" of groups in NIP theories. In particular
we prove the Ellis group conjecture relating the Ellis group to G/G^{00} in
some new cases, including definably amenable groups in o-minimal structures.Comment: 28 pages. Introduction was expanded and some minor mistakes were
corrected. Journal of the London Mathematical Society, accepte
- …