Rigid Differentially Closed Fields
David Marker
本研究借助几何稳定性理论的思路，构建出不具备非平凡自同构的微分闭域，推进微分闭域相关研究。
我们运用几何稳定性理论的思想，构造出无非平凡自同构的微分闭域。
Using ideas from geometric stability theory we construct differentially closed fields with no non-trivial automorphisms.
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A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce-Pillay, is formulated in terms of a relative notion of prolongation for Kolchin-closed sets.
The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that every such field of low degree is isomorphic to a computable differential field. Relativizing the latter result and applying a theorem of Montalban, Soskova, and Soskov, we conclude that the spectra of countable differentially clo…
The class of all countable differentially closed differential fields $K$ of characteristic $0$ was shown by Marker and the author to be "one jump away" from universal for spectra of structures: for every nontrivial countable structure $\mathfrak M$, there is some $K$ whose spectrum is the preimage under jump of the spectrum of $\mathfrak M$, and conversely, for every $K$, there is such an $\mathfrak M$. We show that the missing jump can be accounted for by adding to the signature of differential fields a predicate describing a certain algebraic transcendence property. The ensuing universality …
We give a classification of non-orthogonality classes of trivial order 1 strongly minimal sets in differentially closed fields. A central idea is the introduction of $τ$-forms, functions on the prolongation of a variety which are analogous to 1-forms. Order 1 strongly minimal sets then correspond to smooth projective curves with $τ$-forms. We also introduce $τ$-differentials, algebraic versions of $τ$-forms which are analogous to usual differentials, and develop their basic properties. This enables us to reformulate our classification scheme-theoretically in terms of curves with $τ$-invertible…
The theory of difference-differential fields of characteristic zero has a model-companion denoted by $\it DCFA$. Previously we proved a weak version of Zilber's dichotomy for $\it DCFA$. In this paper we use arc spaces techniques as developed by Moosa, Pillay and Scanlon to suppress the extra hypothesis needed before and prove the full Zilber's dichotomy for $\it DCFA$, we also state how these techniques generalise to partial differential fields with an automorphism.
Separably differentially closed fields
本文引入正特征下的可分微分闭域，证明其为可分闭域的微分类比，并研究其代数与模型论性质，包括基本类、完备化分类、消去量词、稳定性及素模型扩张。
Forking independence in differentially closed fields of positive characteristic
本文在正特征微分闭域理论 DCF$_{p,m}$ 中给出分叉独立性的微分代数刻画，证明实sort代数闭子集上的型平稳，并证明伯努利方程非零解集强极小且几何严格离散，从而在 $\mathbb{F}_p$ 上代数独立。
Let $\mathcal{F}=(F;+,\cdot,0,1,D)$ be a differentially closed field. We consider the question of definability of the derivation $D$ in reducts of $\mathcal{F}$ of the form $\mathcal{F}_{R}=(F;+,\cdot,0,1,P)_{P \in R}$ where $R$ is a collection of definable sets in $\mathcal{F}$. We give examples and non-examples and establish some criteria for definability of $D$. Finally, using the tools developed in the paper we prove that under the assumption of inductiveness of $Th(\mathcal{F}_{R})$ model completeness is a necessary condition for definability of $D$. This can be seen as part of a broader …
Let $(K;+,\cdot, ', 0, 1)$ be a differentially closed field. In this paper we explore the connection between Ax-Schanuel type theorems (predimension inequalities) for a differential equation $E(x,y)$ and the geometry of the set $U:=\{ y:E(t,y) \wedge y' \neq 0 \}$ where $t$ is an element with $t'=1$. We show that certain types of predimension inequalities imply strong minimality and geometric triviality of $U$. Moreover, the induced structure on Cartesian powers of $U$ is given by special subvarieties. If $E$ has some special form then all fibres $U_s:=\{ y:E(s,y) \wedge y' \neq 0 \}$ (with $s…
In the proof of Lemma 2.6 (2) the iteration of the map τ was not performed properly and in fact the lemma is wrong; a counterexample is given by f = \bar{x}_1and k = 2. This error does not, however, affect the geometric characterization given in Theorem 3.4 but only the attempt in Theorem 4.3 to express it as a first-order set of axioms. That attempt is incorrect; the main problem being that in general τV(f_1,..., f_s) 6= V(f_1..., f_s, τf_1,..., τf_s). But a different, indeed simpler, set of first-order axioms, which we will now describe, does express the geometric characterization.
E. Hrushovski proved tha the theory of difference-differential fields has a model companion. We prove this result and other maind properties of this theory that we call DCFA. We describe the SU rank a its relation with transcendence degree. We prove a version of Zilber's dichotomy and we give a description of Abelian definable groups.
Analysability of finite $U$-rank types are explored both in general and in the theory $\mathrm{DCF}_0$. The well-known fact that the equation $δ(\mathrm{log}δx)=0$ is analysable in but not almost internal to the constants is generalized to show that $\underbrace{\mathrm{log}δ...\mathrm{log}δ}_n x=0$ is not analysable in the constants in $(n-1)$-steps. The notion of a \emph{canonical analysis} is introduced -- namely an analysis that is of minimal length and interalgebraic with every other analysis of that length. Not every analysable type admits a canonical analysis. Using properties of reduct…
On ordinary differentially large fields
本文给出了任意特征下单一导子微分大域的Blum公理式刻画，证明特征零下的大域可扩张为微分大域，并应用于证明大域模型的真稠密对理论完备性及闭有序微分域的素模型扩张不存在性。
A natural haystack of differentially closed fields
本文构造了特征0的微分闭域：通过亚纯函数模稠密开集上的等价关系，证明每个极大理想商是微分闭域，并讨论其与力迫、布尔值模型和层的关系。
Rigid Real Closed Fields
本文构造了一个超越维度为二的非阿基米德实闭域，该域具有无非平凡自同构的性质。
Approximate Axiomatization for Differentially-Defined Functions
本文为含 sin、cos、exp 等微分定义函数的实闭域提出完整近似公理化，使真语句可被数值近似证明，并将相关可判定性从闭语句推广到含特殊函数符号的公式。
We prove the Existential Closedness conjecture for the differential equation of the $j$-function and its derivatives. It states that in a differentially closed field certain equations involving the differential equation of the $j$-function have solutions. Its consequences include a complete axiomatisation of $j$-reducts of differentially closed fields, a dichotomy result for strongly minimal sets in those reducts, and a functional analogue of the Modular Zilber-Pink with Derivatives conjecture.
Some More Rigid Real Closed Fields
该研究推广了Marker与Steinhorn的构造方法，得到了所有有限及可数无限超越度的无平凡自同构的可数非阿基米德实闭域，回答了相关问题。
Differentially Large Fields
本文引入微分大域概念，建立其基础理论，给出多种刻画与构造，证明其初等性及在代数扩张下的保持性，表明其在微分域算术中的关键作用。
The theory DCF$_p$A exists for $p>0$
该研究证明特征 $p>0$ 的微分-差分域初等类存在模型伴，即泛型微分自同构下特征 $p$ 微分闭域的理论 DCF$_p$A 存在，同时给出 DCF 与 DCF$_0$A 的替代一阶公理化。
We give an algebro-geometric first-order axiomatization of DCF$_{0,m}$ (the theory of differentially closed fields of characteristic zero with m commuting derivations) in the spirit of the classical geometric axioms of DCF$_0$.
We prove a differential analog of a theorem of Chevalley on extending homomorphisms for rings with commuting derivations, generalizing a theorem of Kac. As a corollary, we establish that, under suitable hypotheses, the image of a differential scheme under a finite morphism is a constructible set. We also obtain a new algebraic characterization of differentially closed fields. We show that similar results hold for differentially closed fields that are saturated, in the sense of model theory. In characteristic p > 0, we obtain related results and establish a differential Nullstellensatz. Analogs…
We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition al…
Embedding stable groups into algebraic groups
该研究在稳定域理论框架下，改编已有证明得出满足可定义闭包条件的类型可定义连通群可嵌入代数群，并给出可定义闭包的统一描述假设，覆盖可分闭域与微分闭域等场景。
Let $\mathcal K=\langle\mathcal R, δ\rangle$ be a closed ordered differential field, in the sense of M. Singer, and $C$ its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal M=\langle \mathcal R, C\rangle$, the $δ$-dimension and the large dimension coincide. As an application, we characterize the definable sets in $\mathcal K$ that are internal to $C$ as those sets that are definable in $\mathcal M$ and have $δ$-dimension $0$. We further show that, for sets definable in $\mathcal K$, having $δ$-dimension $0$ does not generally imply co-analyzability in $…
Finite-dimensional differential-algebraic permutation groups
本文证明了微分闭域中有限秩可定义本原置换群同构于常数域中的代数置换群，验证了相关猜想，并应用于绑定群与微分方程解的超越性。
Assuming that the differential field $(K,δ)$ is differentially large, in the sense of León Sánchez and Tressl, and "bounded" as a field, we prove that for any linear differential algebraic group $G$ over $K$, the differential Galois (or constrained) cohomology set $H^1_δ(K,G)$ is finite. This applies, among other things, to closed ordered differential fields $K$, in the sense of Singer, and to closed $p$-adic differential fields in the sense of Tressl. As an application, we prove a general existence result for parameterized Picard-Vessiot extensions within certain families of fields; if $(K,δ_…
Finding suitably generic points on curves with an application to the construction of rigid real closed fields
该研究针对特征0且超越次数≥2的代数闭域上的不可约曲线，证明了存在代数无关坐标的K-点及对应曲线的此类K-点，进而构造了对应非阿基米德实闭域。
Recent work using the model theory of differentially closed fields to answer questions having to do with the Dixmier-Moeglin equivalence for (noncommutatve) finitely generated noetherian algebras, and for (commutative) finitely generated Poisson algebras, is here surveyed, with an emphasis on the model-theoretic and differential-algebraic-geometric antecedents.
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