Model-complete theories of e-free Ax fields

1983 ◽  
Vol 48 (4) ◽  
pp. 1125-1129
Author(s):  
Moshe Jarden ◽  
William H. Wheeler
Keyword(s):  
Finite Fields ◽  
Complete Theory ◽  
Closed Field ◽  
Easy Consequence ◽  
Interesting Part ◽  
Irreducible Variety ◽  
Free Profinite Group ◽  
Complete Theories ◽  
Separable Closure ◽  
Almost All

This paper's goal is to determine which complete theories of perfect, e-free Ax fields are model-complete. A field K is e-free for a positive integer e if the Galois group g(KS∣K), where Ks is the separable closure of K, is an e-free, profinite group. A perfect field K is pseudo-algebraically closed if each nonvoid, absolutely irreducible variety defined over K has a K-rational point. A perfect, pseudo-algebraically closed field is called an Ax field. The main theorem isA complete theory of e-free Ax fields is model-complete if and only if its field of absolute numbers is e-free.The sufficiency of the latter condition is an easy consequence of a result of Moshe Jarden and Ursel Kiehne [10] and has been noted independently by A. Macintyre and K. McKenna and undoubtedly by others as well. Consequently the necessity of the latter condition is the interesting part of this paper.James Ax [3] initiated the investigation of 1-free Ax fields. He proved that these fields, which he called pseudo-finite fields, are precisely the infinite models of the theory of finite fields. He [3] also presented examples of perfect, 1-free fields which are not pseudo-algebraically closed and an example of a 1-free Ax field whose complete theory is not model-complete. Moshe Jarden [5] showed that the first examples are isolated cases in that almost all, perfect, 1-free, algebraic extensions of a denumerable, Hilbertian field are pseudo-algebraically closed. The results in this paper show that the second example is also an isolated case in that almost all complete theories of 1-free Ax fields are model-complete.

The elementary theory of free pseudo p-adically closed fields of finite corank

1991 ◽  
Vol 56 (2) ◽  
pp. 484-496 ◽  
Author(s):  
Ido Efrat
Keyword(s):  
Rational Point ◽  
Elementary Theory ◽  
Valuation Rings ◽  
Order Language ◽  
Irreducible Variety ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Unary Relation ◽  
Adic Valuation ◽  
Almost All

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

On the Probabilistic Nature of Observable Phenomena

2019 ◽  
Author(s):  
Muhammad Ali
Keyword(s):  
Quantum Mechanics ◽  
Complete Theory ◽  
Interpretation Of Quantum Mechanics ◽  
Multiple Realities ◽  
Lack Of Information ◽  
Complete Theories ◽  
Probabilistic Nature

This paper proposes a Gadenkan experiment named “Observer’s Dilemma”, to investigate the probabilistic nature of observable phenomena. It has been reasoned that probabilistic nature in, otherwise uniquely deterministic phenomena can be introduced due to lack of information of underlying governing laws. Through theoretical consequences of the experiment, concepts of ‘Absolute Complete’ and ‘Observably Complete” theories have been introduced. Furthermore, nature of reality being ‘absolute’ and ‘observable’ have been discussed along with the possibility of multiple realities being true for observer. In addition, certain aspects of quantum mechanics have been interpreted. It has been argued that quantum mechanics is an ‘observably complete’ theory and its nature is to give probabilistic predictions. Lastly, it has been argued that “Everettian - Many world” interpretation of quantum mechanics is very real and true in the framework of ‘observable nature of reality’, for humans.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

Theories with finitely many models

1986 ◽  
Vol 51 (2) ◽  
pp. 374-376 ◽  
Author(s):  
Simon Thomas
Keyword(s):  
Natural Number ◽  
Complete Theory ◽  
Elementary Equivalence ◽  
First Order ◽  
Finite Models ◽  
Simple Observation ◽  
Finite Structure ◽  
Order Language ◽  
Complete Theories

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

Model completeness of o-minimal structures expanded by Dedekind cuts

2005 ◽  
Vol 70 (1) ◽  
pp. 29-60 ◽  
Author(s):  
Marcus Tressl
Keyword(s):  
Valuation Ring ◽  
Ordered Set ◽  
Real Closed Field ◽  
Complete Theory ◽  
Closed Field ◽  
Model Completeness ◽  
Unary Predicate ◽  
Minimal Structure ◽  
Totally Ordered Set ◽  
Dedekind Cuts

§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pL ⋃ pR = M and pL < pR, i.e., a < b for all a ∈ pL, b ∈ pR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L(D)-structure (M, pL) which is model complete.The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {a ∈ M ∣ a > Z} and Z− for the cut q with qL = {a ∈ M ∣ a < Z}.

scholarly journals The asymptotic nature of the analytic spread

1979 ◽  
Vol 86 (1) ◽  
pp. 35-39 ◽  
Author(s):  
M. Brodmann
Keyword(s):  
Local Ring ◽  
Result Section ◽  
Easy Consequence ◽  
Noetherian Local Ring ◽  
Section 8 ◽  
Asymptotic Nature ◽  
Analytic Spread ◽  
Image Position ◽  
Almost All ◽  
Local Nature

In (3), corollary, p. 373) Burch gives the following inequality for the analytic spread l(I) of an ideal I of a noetherian local ring (R, m):In this paper we shall improve this by showing that the number min depth (R/In) may be replaced by the asymptotic value of depth (R/In) for large n (which exists) (see Section (2)). By its definition (see (6), def. 3)) the analytic spread is of asymptotic nature, i.e. depends on the modules In/mIn = Un only for large n. We shall prove a stronger result, Section (4), which also shows the asymptotic nature of l(I). This result might be interesting for itself, particularly as it is not of local nature. Once Section (4) is proved and once we know that depth (R/In) is asymptotically constant (which turns out to be an easy consequence of (1), (1)), our improved inequality is easily established: Indeed, replacing R by R/xR where x is regular with respect to almost all modules (R/In), we perform a change which affects only finitely many of the modules Un (see Section (8)).

scholarly journals ON STABLE QUADRATIC POLYNOMIALS

2012 ◽  
Vol 54 (2) ◽  
pp. 359-369 ◽  
Author(s):  
OMRAN AHMADI ◽  
FLORIAN LUCA ◽  
ALINA OSTAFE ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Fields ◽  
Quadratic Polynomial ◽  
Quadratic Polynomials ◽  
Polynomials Over Finite Fields ◽  
Characteristic 2 ◽  
The Stability ◽  
Almost All

AbstractWe recall that a polynomial f(X) ∈ K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f(X) ∈ ℤ[X] are stable over ℚ. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f(X) ∈ ℤ[X] can be detected by a finite algorithm; this property is closely related to the stability of f(X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.

Decidable regularly closed fields of algebraic numbers

1985 ◽  
Vol 50 (2) ◽  
pp. 468-475 ◽  
Author(s):  
Lou van den Dries ◽  
Rick L. Smith
Keyword(s):  
Galois Group ◽  
Elementary Theory ◽  
Algebraic Closure ◽  
Closed Field ◽  
Algebraic Numbers ◽  
Integer Polynomials ◽  
Forthcoming Book ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Theoretic Argument

A field K is regularly closed if every absolutely irreducible affine variety defined over K has K-rational points. This notion was first isolated by Ax [A] in his work on the elementary theory of finite fields. Later Jarden [J2] and Jarden and Kiehne [JK] extended this in different directions. One of the primary results in this area is that the elementary properties of a regularly closed field K with a free Galois group (on either finitely or countably many generators) are determined by the set of integer polynomials in one indeterminate with a zero in K. The method of proof employed in [J1], [J2] and [JK] is unusual for algebra since it is a measure-theoretic argument. In this brief summary we have not made any attempt at completeness. We refer the reader to the recent paper of Cherlin, van den Dries, and Macintyre [CDM] and to the forthcoming book by Fried and Jarden [FJ] for a more thorough discussion of the latest results. We would like to thank Moshe Jarden, Angus Macintyre, and Zoe Chatzidakis for their comments on an earlier version of this paper.A countable field K is ω-free if the absolute Galois group , where is the algebraic closure of K and is the free profinite group on ℵ0 generators.

scholarly journals Crooked Maps in Finite Fields

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Gohar Kyureghyan
Keyword(s):  
Finite Fields ◽  
Geometrical Characterization ◽  
Power Maps ◽  
International Audience ◽  
Almost All

International audience We consider the maps $f:\mathbb{F}_{2^n} →\mathbb{F}_{2^n}$ with the property that the set $\{ f(x+a)+ f(x): x ∈F_{2^n}\}$ is a hyperplane or a complement of hyperplane for every $a ∈\mathbb{F}_{2^n}^*$. The main goal of the talk is to show that almost all maps $f(x) = Σ_{b ∈B}c_b(x+b)^d$, where $B ⊂\mathbb{F}_{2^n}$ and $Σ_{b ∈B}c_b ≠0$, are not of that type. In particular, the only such power maps have exponents $2^i+2^j$ with $gcd(n, i-j)=1$. We give also a geometrical characterization of this maps.

scholarly journals DISCRETE METRIC SPACES: STRUCTURE, ENUMERATION, AND 0-1 LAWS

2019 ◽  
Vol 84 (4) ◽  
pp. 1293-1325 ◽  
Author(s):  
DHRUV MUBAYI ◽  
CAROLINE TERRY
Keyword(s):  
Metric Spaces ◽  
Extremal Combinatorics ◽  
Binary Relations ◽  
Easy Consequence ◽  
First Order ◽  
Colored Graphs ◽  
Long Line ◽  
Image Position ◽  
Almost All ◽  
Continuous Setting

AbstractFix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots ,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left( {\matrix{ n \cr 2 \cr } } \right) + o\left( {n^2 } \right)} .$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left( {n^2 } \right)$ to $o\left( 1 \right)$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$ . In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots ,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

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