Philip Dittmann, Arno Fehm
We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions.
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We use a generalization of a construction by Ziegler to show that for any field $F$ and any countable collection of countable subsets $A_i \subseteq F, i \in \calI \subset \Z_{>0}$ there exist infinitely many fields $K$ of arbitrary positive transcendence degree over $F$ and of infinite algebraic degree such that each $A_i$ is first-order definable over $K$. We also use the construction to show that many infinitely axiomatizable theories of fields which are not compatible with the theory of algebraically closed fields are finitely hereditarily undecidable.
We provide a sufficient condition for a polynomial ring, not necessarily commutative, to have a first-order definition for the rational integers.
We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of $\mathbb Q$. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for {\it all} extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of ``small'' and ``big'' subrings of number fields…
We extend results of Videla and Fukuzaki to define algebraic integers in large classes of infinite algebraic extensions of Q and use these definitions for some of the fields to show the first-order undecidability. We also obtain a structural sufficient condition for definability of the ring of integers over its field of fractions. In particular, we show that the following propositions hold. (1) For any rational prime $q$ and any positive rational integer $m$, algebraic integers are definable in any Galois extension of Q where the degree of any finite subextension is not divisible by $q^{m}$. (…
We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are (commutative unital) reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $(0,1,+,\cdot)$ . In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type; in this direction, we also provide a separate proof of the undecidability of these rings that works uniformly in any characteristic. These definability and undecidability assertions extend a series of results by Raphae…
This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present two decision algorithms based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures, the second one also providing a formula in the positive case. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical te…
We analyze the effect of replacing several natural uses of definability in set theory by the weaker model-theoretic notion of algebraicity. We find, for example, that the class of hereditarily ordinal algebraic sets is the same as the class of hereditarily ordinal definable sets, that is, HOA = HOD. Moreover, we show that every (pointwise) algebraic model of ZF is actually pointwise definable. Finally, we consider the implicitly constructible universe Imp---an algebraic analogue of the constructible universe---which is obtained by iteratively adding not only the sets that are definable over wh…
We make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired on Serre's algebraic twisting) to describe arbitrary fibres in cohomology sequences -- yielding a useful finiteness result on cohomology sets. Applied to the special case of differential fields and Kolchin's constrained cohomology, we prove that the first constrained cohomology set of a differen…
We deal with various splitting methods in algebraic logic. The word `splitting' refers to splitting some of the atoms in a given relation or cylindric algebra each into one or more subatoms obtaining a bigger algebra, where the number of subatoms obtained after splitting is adjusted for a certain combinatorial purpose. This number (of subatoms) can be an infinite cardinal. The idea originates with Leon Henkin. Splitting methods existing in a scattered form in the literature, possibly under different names, proved useful in obtaining (negative) results on non-atom canonicity, non-finite axiomat…
First-order definability of Campana Points and Darmon Points in algebraic function fields in one variable over number fields
本文基于二次Pfister形式理论，将Koenigsmann在数域中证明Z在Q中普遍可定义的方法推广到代数函数域，从而在更一般的代数簇上定义Campana点和Darmon点。
Definability and decidability for rings of integers in totally imaginary fields
本文证明了全实数域整数环在全虚二次扩张整数环中的存在性可定义性，利用单位群构造全实子集，并在特定JR-数条件下推导出整数环的不可判定性，推广了以往仅适用于相反情形的结论。
Definability and undecidability via the torsion subgroup of units
本文证明整数环在最大阿贝尔扩张的整数环中一阶可定义，从而其理论不可判定，并推广到全实数域加虚数单位的子域，关键是无参数正存在公式定义单位根。
Godel's theory T can be understood as a theory of the simply-typed lambda calculus that is extended to include the constant 0, the successor function S, and the operator R_tau for primitive recursion on objects of type tau. It is known that the functions from non-negative integers to non-negative integers that can be defined in this theory are exactly the <epsilon_0-recursive functions of non-negative integers. As an extension of this result, we show that when the domain and codomain are restricted to pure closed normal forms, the functionals of arbitrary type that are definable in T can be en…
Recently, Anscombe and Koenigsmann gave an existential 0-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields.
We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists in showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loese…
In two papers we noted that in common practice many algebraic constructions are defined only `up to isomorphism' rather than explicitly. We mentioned some questions raised by this fact, and we gave some partial answers. The present paper provides much fuller answers, though some questions remain open. Our main result says that there is a transitive model of Zermelo-Fraenkel set theory with choice (ZFC) in which every fully definable construction is `weakly natural' (a weakening of the notion of a natural transformation). A corollary is that there are models of ZFC in which some well-known cons…
Existential definability and diophantine stability
本文研究数域及其代数扩张的整数环的丢番图定义，在阿贝尔簇秩条件及扩张类型限制下，证明了丢番图稳定性。
We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}_{alg}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in $\mathbb{R}_{an,exp}$ with respect to this structure. As a consequence we reprove that the zero loci of admissible normal functions are algebraic.
A set $G \subseteq ω$ is $n$-generic for a positive integer $n$ if and only if every $Σ^0_n$ formula of $G$ is decided by a finite initial segment of $G$ in the sense of Cohen forcing. It is shown here that every $n$-generic set $G$ is properly $Σ^0_n$ in some $G$-recursive $X$. As a corollary, we also prove that for every $n > 1$ and every $n$-generic set $G$ there exists a $G$-recursive $X$ which is generalized ${\rm low}_n$ but not generalized ${\rm low}_{n-1}$. Thus we confirm two conjectures of Jockusch.
Generalizing a theorem of Campercholi, we characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Isbell, as follows: in any prevariety having at most s non-logical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m + s + aleph0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable "bridge theorems", matching the surjectivity of all epim…
Svenonius theorem reduces the problem of first-order definability to the problem of relationship between groups of permutations. In the present paper we use this approach to describe the lattice of definable relations for the structure of integer numbers with the successor relation.
We study questions around the existence of bounds and the dependence on parameters for linear-algebraic problems in polynomial rings over rings of an arithmetic flavor.In particular, we show that the module of syzygies of polynomials $f_1,...,f_n\in R[X_1,...,X_N]$ with coefficients in a Prüfer domain $R$ can be generated by elements whose degrees are bounded by a number only depending on $N$, $n$ and the degree of the $f_j$. This implies that if $R$ is a Bézout domain, then the generators can be parametrized in terms of the coefficients of $f_1,...,f_n$ using the ring operations and a certain…
First-order definability of Darmon points in number fields
本文研究了数域中Darmon点的一阶可定义性，通过全称存在全称公式描述，并在特定情况下改进为全称存在定义，同时展示了公式的统一性和多项式次数的界。
We prove that infinite p-adically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree-2 extension of a totally real number field, then there exists a prime p of K and a set of K-primes S of density arbitrarily close to 1 such that there is an infinite p-adically discrete set that is Diophantine over the ring O_{K,S} of S-integers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of K-primes S of density 1 and an infinite Diophant…
On a lattice of relational spaces (reducts) for the order of integers
该研究聚焦整数序的可定义性格，描述由七种特定关系生成的子格结构，为整数序关系空间的约化研究提供基础，同时提出开放问题。
First-order definitions of rings of integral functions over algebraic extensions of function fields and undecidability
本文研究$\u211d_p(t)$的$q$-有界无限代数扩张及其整函数环的一阶可定义性与不可判定性，利用范数方程与哈塞范数原理得到整闭包可定义性、一阶理论不可判定等更强结果。
Definability is a key notion in the theory of Grothendieck fibrations that characterises when an external property of objects can be accessed from within the internal logic of the base of a fibration. In this paper we consider a generalisation of definability from properties of objects to structures on objects, introduced by Shulman under the name local representability. We first develop some general theory and show how to recover existing notions due to Bénabou and Johnstone as special cases. We give several examples of definable and non definable notions o structure, focusing on algebraic we…
We study the model theory of the $2$-sorted structure $(\mathbb{F}, \mathbb{C};χ)$, where $\mathbb{F}$ is an algebraic closure of a finite field of characteristic $p$, $\mathbb{C}$ is the field of complex numbers and $χ: \mathbb{F} \to \mathbb{C}$ is an injective, multiplication preserving map. We obtain an axiomatization $\mathrm{ACFC}_p$ of $\mathrm{Th}(\mathbb{F},\mathbb{C};χ)$ in a suitable language $L$, classify the models of $\mathrm{ACFC}_p$ up to isomorphism, prove a modified model companion result, give various descriptions of definable sets inside a model of $\mathrm{ACFC}_p$, and de…
Ehrenfeucht's lemma (1973) asserts that whenever one element of a model of Peano arithmetic is definable from another, then they satisfy different types. We consider here the analogue of Ehrenfeucht's lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and in particular, Ehrenfeucht's lemma holds fully for models of set theory satisfying $V=HOD$. We show that the lemma can fail, however, in models of set theory with $V\neq HOD$, and it necessarily fails in the forcing extension to add a generic Cohen real. We go o…
Notes on Algebraic Properties and Non-Standard Analysis of the Ring of Integers Modulo Infinitely Large Primes
本文探讨无限大素数模整数环的代数与模型论性质，结合非标准分析方法扩展超越数理论中的判别准则。
We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for differential algebraic groups. The theorem establishes a sufficient criterion for the subgroup of a differential algebraic group generated by an infinite family of subvarieties to be a differential algebraic subgroup. This theorem is used for various definability results. For instance, we show every noncommutative almost simple differential algebraic group is perf…
Undecidability of infinite algebraic extensions of $\mathbb{F}_p(t)$
本文建立一般框架研究 $\mathbb{F}_p(t)$ 的无限代数扩张的可定义性与可判定性，并利用特征标理论构造非等平凡椭圆曲线族，证明对每个奇素数 $p$ 存在无穷多个素数 $r$ 使相应域的一阶理论不可判定。
Defining $\mathbb Z$ using unit groups
本文研究 $\Q$ 代数扩张整数环上的一阶可定义性与可判定性问题，通过单位群给出 $\Z$ 的简洁一致定义，并据此证明了可构造数域的不可判定性。
We prove that the theory of all modules over the ring of algebraic integers is decidable.
We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p-valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in CODF, we establish a relative Galois correspondence for relatively definable subgroups of the group of differential order automorp…
Definable Eventual Equalizers
研究在特定值微分域（如超序列微分域）中代数微分方程解的牛顿图方法，证明最终等化子可从输入微分多项式系数一致且可定义地获得，并得到类似的可定义性结果。
We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented, while maintaining its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring, in which we can formulate a form of integration resembling Kontsevich's motivic integration via arc schemes. Whereas the original construction was via definability, we have translated in this paper everything into a topos-theoretic framework.
In this paper we develop a theory called $σ$-Set Theory, in which we present an axiom system developed from the study of Set Theories of Zermelo-Fraenkel, Neumann-Bernays-Godel and Morse-Kelley. In $σ$-Set Theory, we present the proper existence of objects called $σ$-antielement, $σ$-antiset, natural numbers, antinatural numbers and generated $σ$-set by two $σ$-sets, from which we obtain, among other things, a commutative non-associative algebraic structure called Integer Space $3^{X}$, which corresponds to the algebraic completion of $2^{X}$.
On Some Systems of Equations in Abelian Varieties
本文证明了 Bays 和 Kirby 提出的阿贝尔指数-代数闭性猜想的一个情形，即对切丛中分裂为李代数线性子空间与代数簇乘积的子簇成立，运用同调、微分拓扑和 o-极小性技术给出自包含证明。
We prove decidability of univariate real algebra extended with predicates for rational and integer powers, i.e., $(x^n \in \mathbb{Q})$ and $(x^n \in \mathbb{Z})$. Our decision procedure combines computation over real algebraic cells with the rational root theorem and witness construction via algebraic number density arguments.
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