19 research outputs found
On n-dependent groups and fields
First, an example of a 2-dependent group without a minimal subgroup of
bounded index is given. Second, all infinite n-dependent fields are shown to be
Artin-Schreier closed. Furthermore, the theory of any non separably closed PAC
field has the IPn property for all natural numbers n and certain properties of
dependent (NIP) valued fields extend to the n-dependent context
Mekler's construction and generalized stability
Mekler's construction gives an interpretation of any structure in a finite
relational language in a group (nilpotent of class and exponent , but
not finitely generated in general). Even though this construction is not a
bi-interpretation, it is known to preserve some model-theoretic tameness
properties of the original structure including stability and simplicity. We
demonstrate that -dependence of the theory is preserved, for all , and that NTP is preserved. We apply this result to obtain
first examples of strictly -dependent groups (with no additional structure).Comment: v.2 many minor corrections and presentation improvements throughout
the article, more details were added in some of the proofs; Remarks 2.12,
2.13 and Problem 5.8 are new; accepted to the Israel Journal of Mathematic
An improved bound for regular decompositions of -uniform hypergraphs of bounded -dimension
A regular partition for a -uniform hypergraph
consists of a partition and for each , a partition , such that certain quasirandomness properties hold. The
complexity of is the pair . In this paper we show that
if a -uniform hypergraph has -dimension at most , then there is
a regular partition for of complexity , where
is bounded by a polynomial in the degree of regularity. This is a vast
improvement on the bound arising from the proof of this regularity lemma in
general, in which the bound generated for is of Wowzer type. This can be
seen as a higher arity analogue of the efficient regularity lemmas for graphs
and hypergraphs of bounded VC-dimension due to Alon-Fischer-Newman,
Lov\'{a}sz-Szegedy, and Fox-Pach-Suk
Model-theoretic properties of nilpotent groups and Lie algebras
We give a systematic study of the model theory of generic nilpotent groups
and Lie algebras. We show that the Fra\"iss\'e limit of 2-nilpotent groups of
exponent studied by Baudisch is 2-dependent and NSOP. We prove that
the class of -nilpotent Lie algebras over an arbitrary field, in a language
with predicates for a Lazard series, is closed under free amalgamation. We show
that for , the generic -nilpotent Lie algebra over
is strictly NSOP and -dependent. Via the Lazard correspondence, we
obtain the same result for -nilpotent groups of exponent , for an odd
prime
NIPn CHIPS
We give general conditions under which classes of valued fields have NIPn
transfer and generalize the Anscombe-Jahnke classification of NIP henselian
valued fields to NIPn henselian valued fields.Comment: 17 page
, and division rings of prime characteristic
Combining a characterisation by Bélair, Kaplan, Scanlon and Wagner of certain valued fields of characteristic with Dickson's construction of cyclic algebras, we provide examples of noncommutative division ring of characteristic and show that an division ring of characteristic has finite dimension over its centre, in the spirit of Kaplan and Scanlon's proof that infinite fields have no Artin-Schreier extension. The result extends to division rings of characteristic , using results of Chernikov, Kaplan and Simon. We also highlight consequences of our proofs that concern or simple difference fields
Sets, groups, and fields definable in vector spaces with a bilinear form
We study definable sets, groups, and fields in the theory of
infinite-dimensional vector spaces over an algebraically closed field equipped
with a nondegenerate symmetric (or alternating) bilinear form. First, we define
an ()-valued dimension on definable
sets in enjoying many properties of Morley rank in strongly minimal
theories. Then, using this dimension notion as the main tool, we prove that all
groups definable in are (algebraic-by-abelian)-by-algebraic, which,
in particular, answers a question of Granger. We conclude that every infinite
field definable in is definably isomorphic to the field of scalars
of the vector space. We derive some other consequences of good behaviour of the
dimension in , e.g. every generic type in any definable set is a
definable type; every set is an extension base; every definable group has a
definable connected component.
We also consider the theory of vector spaces over a real
closed field equipped with a nondegenerate alternating bilinear form or a
nondegenerate symmetric positive-definite bilinear form. Using the same
construction as in the case of , we define a dimension on sets
definable in , and using it we prove analogous results about
definable groups and fields: every group definable in is
(semialgebraic-by-abelian)-by-semialgebraic (in particular, it is
(Lie-by-abelian)-by-Lie), and every field definable in is
definable in the field of scalars, hence it is either real closed or
algebraically closed.Comment: v2: The particular bounds on dimension obtained in Section 3 were
corrected, and a number of minor corrections has been made throughout the
pape