15 research outputs found
Fields and rings with few types
Let R be an associative ring with possible extra structure. R is said to be
weakly small if there are countably many 1-types over any finite subset of R.
It is locally P if the algebraic closure of any finite subset of R has property
P. It is shown here that a field extension of finite degree of a weakly small
field either is a finite field or has no Artin-Schreier extension. A weakly
small field of characteristic 2 is finite or algebraically closed. Every weakly
small division ring of positive characteristic is locally finite dimensional
over its centre. The Jacobson radical of a weakly small ring is locally
nilpotent. Every weakly small division ring is locally, modulo its Jacobson
radical, isomorphic to a product of finitely many matrix rings over division
rings
Amalgamation of types in pseudo-algebraically closed fields and applications
This paper studies unbounded PAC fields and shows an amalgamation result for
types over algebraically closed sets. It discusses various applications, for
instance that omega-free PAC fields have the property NSOP3. It also contains a
description of imaginaries in PAC fields.Comment: Minor changes in v3. Accepted versio
Randomizations of models as metric structures
The notion of a randomization of a first order structure was introduced by
Keisler in the paper Randomizing a Model, Advances in Math. 1999. The idea was
to form a new structure whose elements are random elements of the original
first order structure. In this paper we treat randomizations as continuous
structures in the sense of Ben Yaacov and Usvyatsov. In this setting, the
earlier results show that the randomization of a complete first order theory is
a complete theory in continuous logic that admits elimination of quantifiers
and has a natural set of axioms. We show that the randomization operation
preserves the properties of being omega-categorical, omega-stable, and stable
Model theory of partially random structures
Many interactions between mathematical objects, e.g. the interaction between the set of primes and the additive structure of N, can be usefully thought of as random modulo some obvious obstructions. In the first part of this thesis, we document several such situations, show that the randomness in these interactions can be captured using first-order logic, and deduce in consequence many model-theoretic properties of the corresponding structures. The second part of this thesis develops a framework to study the aforementioned situations uniformly, shows that many examples of interest in model theory fit into this framework, and recovers many known model-theoretic phenomena about these examples from our results
Specialization of Difference Equations and High Frobenius Powers
We study valued fields equipped with an automorphism which is
locally infinitely contracting in the sense that for
all . We show that various notions of valuation theory, such
as Henselian and strictly Henselian hulls, admit meaningful transformal
analogues. We prove canonical amalgamation results, and exhibit the way that
transformal wild ramification is controlled by torsors over generalized vector
groups. Model theoretically, we determine the model companion: it is decidable,
admits a simple axiomatization, and enjoys elimination of quantifiers up to
algebraically bounded quantifiers.
The model companion is shown to agree with the limit theory of the Frobenius
action on an algebraically closed and nontrivially valued field. This opens the
way to a motivic intersection theory for difference varieties that was
previously available only in characteristic zero. As a first consequence, the
class of algebraically closed valued fields equipped with a distinguished
Frobenius is decidable, uniformly in .Comment: identical to v1 apart from slight modifications in abstrac
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Ordered geometry in Hilbertâs Grundlagen der Geometrie
The Grundlagen der Geometrie brought Euclidâs ancient axioms up to the standards
of modern logic, anticipating a completely mechanical verification of their theorems.
There are five groups of axioms, each focused on a logical feature of Euclidean geometry.
The first two groups give us ordered geometry, a highly limited setting where
there is no talk of measure or angle. From these, we mechanically verify the Polygonal
Jordan Curve Theorem, a result of much generality given the setting, and subtle
enough to warrant a full verification.
Along the way, we describe and implement a general-purpose algebraic language
for proof search, which we use to automate arguments from the first axiom group. We
then follow Hilbert through the preliminary definitions and theorems that lead up to
his statement of the Polygonal Jordan Curve Theorem. These, once formalised and
verified, give us a final piece of automation. Suitably armed, we can then tackle the
main theorem