arXiv subjects
Publications and source records attributed to Elliot Kaplan.
We show that large rosy fields are bounded, and substantially simplify the proofs that large stable fields are separably closed and that large simple fields are bounded. Our proofs go through in a general topological setting. We obtain instability, non-simplicity, and non-rosiness explicitly by building the appropriate trees of definable sets. We also show that orders on large rosy fields have several properties of orders on pseudo real closed fields.
In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical. Combining the first equivalence with an earlier result of Hrushovski, we deduce that algebraically bounded stable fields are precisely expansions of algebraically closed fields by constants. We then consider algebraically bounded and o-minimal expansions of fields with generic derivations. We show that if $\mathbb{M}$ is a simple algebraically bounded structure and $\Delta$ is a generic tuple of derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is supersimple if and only if the derivations commute. Similarly, if $\mathbb{M}$ is an o-minimal structure and $\Delta$ is a generic tuple of $T$-derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is superrosy if and only if the derivations commute. We obtain explicit bounds on ranks using the Kolchin polynomial.
We study the automorphism group of the field of surreal numbers. Our main structure theorem presents a decomposition of this group into a product of five significant factors. Using the representation of surreal numbers as generalized power series via their Conway normal form, we apply results on Hahn fields and groups from the literature in order to obtain this decomposition. Moreover, we provide explicit descriptions of the individual factors enabling us to construct automorphisms on the field of surreal numbers from simpler components. We then extend our study to strongly linear automorphisms in connection to derivations, as well as automorphisms that preserve further exponential structure on the surreals.
We compare Fornasiero and Terzo's framework of generic derivations on algebraically bounded structures with Le\'on S\'anchez and Tressl's differentially large fields. We show in the case of a single derivation that genericity and differential largeness coincide for \'ez-fields, as introduced by Walsberg and Ye. We also show that an NTP$_2$ algebraically bounded structure remains NTP$_2$ after expanding by a generic derivation.
In this paper we completely characterize all dimension functions on all models of the theory $T_{\log}$ of the asymptotic couple of the field of logarithmic transseries (Dimension Theorem). This is done by characterizing the "small" $1$-variable definable sets (Small Sets Theorem). As a byproduct, we show that $T_{\log}$ is d-minimal and does not eliminate imaginaries. Separately, we provide an abstract criterion for d-minimality, which we use to observe some new examples of d-minimal expansions of valued fields.
Let $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We consider the expansion of $\mathbb{T}$ by the binary map that sends a real number $r$ and a positive transseries $f$ to the transseries $f^r$. Building on recent work of Aschenbrenner, van den Dries, and van der Hoeven, we show that this expansion is model complete, and we give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. We show that maximal Hardy fields, equipped with the same map $(f,r)\mapsto f^r$, enjoy the same theory as $\mathbb{T}$, and we use this to establish a transfer theorem between Hardy fields and transseries.
Let $T$ be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that $T$ is power bounded. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring $\mathcal{O}$ and a $T$-derivation $\partial$ such that $\partial$ is monotone, i.e., weakly contractive with respect to the valuation induced by $\mathcal{O}$. We show that the theory of monotone $T$-convex $T$-differential fields, i.e., the common theory of such $K$, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call $T^{\partial}$-henselianity. We establish an Ax--Kochen/Ershov theorem and further results for monotone $T$-convex $T$-differential fields that are $T^{\partial}$-henselian.
We prove a dichotomy for o-minimal fields $\mathcal{R}$, expanded by a $T$-convex valuation ring (where $T$ is the theory of $\mathcal{R}$) and a compatible monomial group. We show that if $T$ is power bounded, then this expansion of $\mathcal{R}$ is model complete (assuming that $T$ is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if $\mathcal{R}$ defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model theoretic tameness.
Let $T$ be a polynomially bounded o-minimal theory extending the theory of real closed ordered fields. Let $K$ be a model of $T$ equipped with a $T$-convex valuation ring and a $T$-derivation. If this derivation is continuous with respect to the valuation topology, then we call $K$ a $T$-convex $T$-differential field. We show that every $T$-convex $T$-differential field has an immediate strict $T$-convex $T$-differential field extension which is spherically complete. In some important cases, the assumption of polynomial boundedness can be relaxed to power boundedness.
We consider a tuple $\Phi = (\phi_1,\ldots,\phi_m)$ of commuting maps on a finitary matroid $X$. We show that if $\Phi$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{\phi_1^{r_1}\cdots\phi_m^{r_m}(a):a \in A\text{ and }r_1+\cdots+r_m = t\}$ is eventually a polynomial in $t$ (we also give a multivariate version of the polynomial). This allows us easily recover Khovanskii's theorem on the growth of sumsets, the existence of the classical Hilbert polynomial, and the existence of the Kolchin polynomial. We also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as a new result on the growth of Betti numbers in simplicial complexes.
Let $T$ be an o-minimal theory extending the theory of real closed ordered fields. An $H_T$-field is a model $K$ of $T$ equipped with a $T$-derivation such that the underlying ordered differential field of $K$ is an $H$-field. We study $H_T$-fields and their extensions. Our main result is that if $T$ is power bounded, then every $H_T$-field $K$ has either exactly one or exactly two minimal Liouville closed $H_T$-field extensions up to $K$-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as $T = \operatorname{Th}(\mathbb{R}_{\operatorname{an},\exp})$.
In [26], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to [15], piggybacking on the just-said results, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling's conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $\mathbf{No}$, which includes $\mathbf{No}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field $\mathbb{T}^{LE}$ of logarithmic-exponential transseries into $\mathbf{No}$ is shown to be initial, as are the ordered exponential fields $\mathbb{R}((ω))^{EL}$ and $\mathbb{R}\langle\langleω\rangle \rangle$.
Let $\mathbb{T}$ be the differential field of logarithmic-exponential transseries. We show that the expansion of $\mathbb{T}$ by its natural exponential function is model complete and locally o-minimal. We give an axiomatization of the theory of this expansion that is effective relative to the theory of the real exponential field. We adapt our results to show that the expansion of $\mathbb{T}$ by this exponential function and by its natural restricted sine and restricted cosine functions is also model complete and locally o-minimal.
Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function $f:[0,1] \to [0,1]$ is $r$-regular if there is a Büchi automaton that accepts precisely the set of base $r \in \mathbb{N}$ representations of elements of the graph of $f$. We show that a continuous $r$-regular function $f$ is locally affine away from a nowhere dense, Lebesgue null, subset of $[0,1]$. As a corollary we establish that every differentiable $r$-regular function is affine. It follows that checking whether an $r$-regular function is differentiable is in $\operatorname{PSPACE}$. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo.
This paper proposes a new setup for studying pairs of structures. This new framework includes many of the previously studied classes of pairs, such as dense pairs of o-minimal structures, lovely pairs, fields with Mann groups, and $H$-structures, but also includes new ones, such as pairs consisting of a real closed field and a pseudo real closed subfield, and pairs of vector spaces with different fields of scalars. We use the larger generality of this framework to answer three concrete open questions raised in earlier work on this subject.
Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $\delta$ on models $\mathcal{M}\models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(\emptyset)$-definable $\mathcal{C}^1$-functions on $\mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^\delta_G$. The derivation in models $(\mathcal{M},\delta)\models T^\delta_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $\mathcal{M}$. If $T =$ RCF, then $T^\delta_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^\delta_G$ has $T$ as its open core, that $T^\delta_G$ is distal, and that $T^\delta_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.
We define the field $\mathbb{L}$ of logarithmic hyperseries, construct on $\mathbb{L}$ natural operations of differentiation, integration, and composition, establish the basic properties of these operations, and characterize these operations uniquely by such properties.
We show that the theory $T_{\log}$ of the asymptotic couple of the field of logarithmic transseries is distal. As distal theories are NIP (= the non-independence property), this provides a new proof that $T_{\log}$ is NIP. Finally, we show that $T_{\log}$ is not strongly dependent, and in particular, it is not $\operatorname{dp}$-minimal and it does not have finite $\operatorname{dp}$-rank.