scholarly journals Pseudofinite difference fields

2019 ◽  
Vol 19 (02) ◽  
pp. 1950011
Author(s):  
Tingxiang Zou
Keyword(s):  
Finite Fields ◽  
Transcendence Degree ◽  
Order Property ◽  
Frobenius Automorphism ◽  
Difference Fields ◽  
Definable Sets ◽  
Partial Connection

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we establish a partial connection between coarse dimension and transformal transcendence degree in these difference fields.

scholarly journals Pseudofinite difference fields and counting dimensions

2020 ◽  
Vol 21 (01) ◽  
pp. 2050022
Author(s):  
Tingxiang Zou
Keyword(s):  
Finite Fields ◽  
Transcendence Degree ◽  
Order Property ◽  
Frobenius Automorphism ◽  
Difference Fields ◽  
Definable Sets

We study a family of ultraproducts of finite fields with the Frobenius automorphism in this paper. Their theories have the strict order property and TP2. But the coarse pseudofinite dimension of the definable sets is definable and integer-valued. Moreover, we also discuss the possible connection between coarse dimension and transformal transcendence degree in these difference fields.

Effective counting of the points of definable sets over finite fields

1994 ◽  
Vol 85 (1-3) ◽  
pp. 103-133 ◽  
Author(s):  
Michael D. Fried ◽  
Dan Haran ◽  
Moshe Jarden
Keyword(s):  
Finite Fields ◽  
Definable Sets

scholarly journals Motives for perfect PAC fields with pro-cyclic Galois group

2008 ◽  
Vol 73 (3) ◽  
pp. 1036-1050
Author(s):  
Immanuel Halupczok
Keyword(s):  
Galois Group ◽  
Finite Fields ◽  
Grothendieck Ring ◽  
Additional Information ◽  
Chow Motives ◽  
Closed Fields ◽  
Cohomological Invariant ◽  
Definable Sets ◽  
Grothendieck Rings ◽  
Algebraically Closed Fields

AbstractDenef and Loeser denned a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group.In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective.

Asymptotic classes of finite structures

2007 ◽  
Vol 72 (2) ◽  
pp. 418-438 ◽  
Author(s):  
Richard Elwes
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Good Control ◽  
First Order ◽  
Finite Structures ◽  
Definable Sets

In this paper we consider classes of finite structures where we have good control over the sizes of the definable sets. The motivating example is the class of finite fields: it was shown in [1] that for any formula in the language of rings, there are finitely many pairs (d, μ) ∈ ω × Q>0 so that in any finite field F and for any ā ∈ Fm the size |ø(Fn,ā)| is “approximately” μ|F|d. Essentially this is a generalisation of the classical Lang-Weil estimates from the category of varieties to that of the first-order-definable sets.

scholarly journals Pseudofinite Structures and Counting Dimensions

2021 ◽  
Vol 27 (2) ◽  
pp. 223-223
Author(s):  
Tingxiang Zou
Keyword(s):  
Classical Model ◽  
Explicit Construction ◽  
Permutation Groups ◽  
Minimal Set ◽  
Existing Structures ◽  
Finite Structures ◽  
Difference Fields ◽  
Abstract Notion ◽  
Partial Connection ◽  
E Mail

AbstractThe thesis pseudofinite structures and counting dimensions is about the model theory of pseudofinite structures with the focus on groups and fields. The aim is to deepen our understanding of how pseudofinite counting dimensions can interact with the algebraic properties of underlying structures and how we could classify certain classes of structures according to their counting dimensions. Our approach is by studying examples. We treat three classes of structures: The first one is the class of H-structures, which are generic expansions of existing structures. We give an explicit construction of pseudofinite H-structures as ultraproducts of finite structures. The second one is the class of finite difference fields. We study properties of coarse pseudofinite dimension in this class, show that it is definable and integer-valued and build a partial connection between this dimension and transformal transcendence degree. The third example is the class of pseudofinite primitive permutation groups. We generalise Hrushovski’s classical classification theorem for stable permutation groups acting on a strongly minimal set to the case where there exists an abstract notion of dimension, which includes both the classical model theoretic ranks and pseudofinite counting dimensions. In this thesis, we also generalise Schlichting’s theorem for groups to the case of approximate subgroups with a notion of commensurability.Abstract prepared by Tingxiang Zou.E-mail: [email protected]: https://tel.archives-ouvertes.fr/tel-02283810/document

ON DIFFERENCE FIELDS WITH QUANTIFIER ELIMINATION

2001 ◽  
Vol 33 (6) ◽  
pp. 641-646 ◽  
Author(s):  
RAHIM MOOSA
Keyword(s):  
Quantifier Elimination ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Integer Power ◽  
Frobenius Automorphism ◽  
Difference Field ◽  
Difference Fields

This paper proves that a difference field (E, σ) admits quantifier elimination if and only if E is an algebraically closed field, and σ is an integer power of the Frobenius automorphism.

Finite Fields

1996 ◽  
Author(s):  
Rudolf Lidl ◽  
Harald Niederreiter
Keyword(s):  
Finite Fields
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