9 research outputs found
On definable -generic groups and minimal flows in -adically closed fields
Let be a definable group definable over a small model . Recall that
a global type on is definable -generic over if every left
translate of is definable over . We call strongly -generic over
if every left translate of does not fork over .
Let be a group definable over the field of -adic
numbers admitting global definable -generic types over . We
show that has unboundedly many global weakly generic types iff there is a
global type on which is strongly -generic over and a
-definable function such that is finitely
satisfiable in .
Recall that the -type on is the partial type consisting of
the formulas over which define open neighborhoods of the
identity of . We show that every global weakly generic type on is
-invariant: For any and , we have
.
Let be groups definable over such that is a normal
subgroup of and is a definably compact group. Then we show that the
weakly generic types on coincide with almost periodic types iff has
boundedly many global weakly generic types
Constructive aspects of Kochen\u27s theorem on p-adic closures
In this work we begin with a brief survey of set theory and arithmetic to provide background for a logical procedure to `cleanse\u27 the Axiom of Choice from a proof of a theorem of Kochen\u27s. We accomplish this in the following chapters. We then discuss certain theorems involving definable Skolem functions. These theorems are used in Chapter 5 to give a construction of a p-adic closure of a p-valued field. Certain further considerations and open questions are addressed in the _x000C_final chapter
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
So
Model theory of finite and pseudofinite rings
The model theory of finite and pseudofinite fields as well as the model theory of finite and
pseudofinite groups have been and are thoroughly studied. A close relation has been found
between algebraic and model theoretic properties of pseudofinite fields and psedudofinite
groups.
In this thesis we present results contributing to the beginning of the study of model
theory of finite and pseudofinite rings.
In particular we classify the theory of ultraproducts of finite residue rings in the context
of generalised stability theory. We give sufficient and necessary conditions for the theory
of such ultraproducts to be NIP, simple, NTP2 but not simple nor NIP, or TP2 .
Further, we show that for any fixed positive l in N the class of finite residue rings
{Zp=p^l Zp : p in P} forms an l-dimensional asymptotic class. We discuss related classes
of finite residue rings in the context of R-multidimensional asymptotic classes.
Finally we present a classification of simple and semisimple (in the algebraic sense)
pseudofinite rings, we study NTP2 classes of J-semisimple rings and we discuss NIP
classes of finite rings and ultraproducts of these NIP classes
S-parts of values of univariate polynomials, binary forms and decomposable forms at integral points
Let be a finite set of primes. The -part of a non-zero integer
is the largest positive divisor of that is composed of primes from .
In 2013, Gross and Vincent proved that if is a polynomial with integer
coefficients and with at least two roots in the complex numbers, then for every
integer at which is non-zero, we have (*) , where and are effectively computable and . Their proof
uses Baker-type estimates for linear forms in complex logarithms of algebraic
numbers. As an easy application of the -adic Thue-Siegel-Roth theorem we
show that if has degree and no multiple roots, then an
inequality such as (*) holds for all , provided we do not require
effectivity of . Further, we show that such an inequality does not hold
anymore with and sufficiently small . In addition we prove a density
result, giving for every an asymptotic estimate with the right
order of magnitude for the number of integers with absolute value at most
such that has -part at least . The result of
Gross and Vincent, as well as the other results mentioned above, are
generalized to values of binary forms and decomposable forms at integral
points. Our main tools are Baker type estimates for linear forms in complex and
-adic logarithms, the -adic Subspace Theorem of Schmidt and Schlickewei,
and a recent general lattice point counting result of Barroero and Widmer.Comment: 42 page
Definable equivalence relations and zeta functions of groups
The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.Peer reviewedPostprin
Non-archimedean stratifications in T-convex fields.
We prove that whenever T is a power-bounded o-minimal theory, t-stratifications exist for definable maps and sets in T-convex fields. To this effect, a thorough analysis of definability in T-convex fields is carried out. One of the conditions required for the result above is the Jacobian property, whose proof in this work is a long and technical argument based on an earlier proof of this property for valued fields with analytic structure. An example is given to illustrate that t-stratifications do not exist in general when T is not power-bounded. We also show that if T is power-bounded, the theory of all T-convex fields is b-minimal with centres.
We also address several applications of tstratifications. For this we exclusively work with a power-bounded T. The first application establishes that a t-stratification of a definable set X in a T-convex field induces t stratifications on the tangent cones of X. This is a contribution to local geometry and singularity theory. Regarding R as a model of T, the remaining applications are derived by considering the stratifications induced on R by t-stratifications in non-standard models. We prove that each such induced stratification is a C1-Whitney stratification; this in turn leads to a new proof of the existence of Whitney stratifications for definable sets in R. We also deal with interactions between tangent cones of definable sets in R and stratifications
Distribution of orders in number fields
In this paper, we study the distribution of orders of bounded discriminants in number fields. We use the zeta functions introduced by Grunewald, Segal, and Smith. In order to carry out our study, we use p-adic and motivic integration techniques to analyze the zeta function. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field. We also obtain non-trivial bounds for higher degree number fields
On The Applications of Lifting Techniques
Lifting techniques are some of the main tools in solving a variety of different computational problems related to the field of computer algebra. In this thesis, we will consider two fundamental problems in the fields of computational algebraic geometry and number theory, trying to find more efficient algorithms to solve such problems.
The first problem, solving systems of polynomial equations, is one of the most fundamental problems in the field of computational algebraic geometry. In this thesis, We discuss how to solve bivariate polynomial systems over either k(T ) or Q using a combination of lifting and modular composition techniques. We will show that one can find an equiprojectable decomposition of a bivariate polynomial system in a better time complexity than the best known algorithms in the field, both in theory and practice.
The second problem, polynomial factorization over number fields, is one of the oldest problems in number theory. It has lots of applications in many other related problems and there have been lots of attempts to solve the problem efficiently, at least, in practice. Finding p-adic factors of a univariate polynomial over a number field uses lifting techniques. Improving this step can reduce the total running time of the factorization in practice. We first introduce a multivariate version of the Belabas factorization algorithm over number fields. Then we will compare the running time complexity of the factorization problem using two different representations of a number field, univariate vs multivariate, and at the end as an application, we will show the improvement gained in computing the splitting fields of a univariate polynomial over rational field