Definable types in the theory of closed ordered differential fields

We study definable types in the theory of closed ordered differential fields (CODF). We show a condition for a type to be definable, then we prove that definable types are dense in the Stone space of CODF.     

Definability of the ring of integers in pro-p galois extensions of number fields

For some infinite algebraic extensionsM of the field of rational numbers we show that the integral closure of ℤ is first-order definable. As an application we prove that the Pythagorean hull of ℚ is an undecidable field.     

Upper bounds for the number of orbital topological types of planar polynomial vector fields modulo limit cycles

The purpose of this paper is to find an upper bound for the number of orbital topological types of nth-degree polynomial fields in the plane. An obstacle to obtaining such a bound is related to the unsolved second part of the Hilbert 16th problem. This obstacle is avoided by introducing the notion of equivalence modulo limit cycles. Earlier, the author obtained a lower bound of the form $2^{cn^2 } $ . In the present paper, an upper bound of the same form but with a different constant is found. Moreover, for each planar polynomial vector field with finitely many singular points, a marked planar graph is constructed that represents a complete orbital topological invariant of this field.     

Abelian-by-central Galois groups of fields II: Definability of inertia/decomposition groups

This paper explores some first-order properties of commuting-liftable pairs in pro-? abelian-by-central Galois groups of fields. The main focus of the paper is to prove that minimized inertia and decomposition groups of many valuations are first-order definable using a predicate for the collection of commuting-liftable pairs. For higher-dimensional function fields over algebraically closed fields, we show that the minimized inertia and decomposition groups of quasi-divisorial valuations are uniformly first-order definable in this language.     

Polynomials in topological vector spaces over valued fields

We consider the vector space of continuousm-homogeneous polynomials between topological vector spaces over a non-trivially valued field of characteristic zero and certain natural vector topologies on such spaces, and we prove polynomial versions of certain well known theorems of the linear theory of locally convex spaces. Partially supported by CNPq.     

On topological types of ordered median functions

An ordered median functions is a continuous piecewise-linear function. It is well known that in finite dimensional spaces every continuous piecewise-linear function admits a max-min representation in terms of its linear functions. An explicit representation of an ordered median function in max-min form is given by the authors and will appear in a forthcoming issue of this journal. Based on this representation, we give a topological classification of ordered median functions through their simplicial complex of ascent (resp. descent) cones.     

Definability and automorphisms in abstract logics

In any model theoretic logic, Beths definability property together with Feferman-Vaughts uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craigs interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowskis theorem. Adding compactness, we obtain the full version of this theorem. Various combinations of definability and uniform reduction relative to other logics yield corresponding results on the existence of non-rigid models.     

On a characterization of path connected topological fields

The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space X and the space of maximal ideals of the ring of real valued continuous functions $C(X,\mathbb{R})$. More explicitly, our motivation is the following question: What is the essential property of the topological field $F=\mathbb{R}$ that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if F is a path connected topological field.     

Ordinary differential inequalities and quasimonotonicity in ordered topological vector spaces

A well known comparison theorem on ordinary differential inequalities with quasimonotone right-hand side was carried over by
Volkmann (1972) to (pre)ordered topological vector spaces. We prove that the quasimonotonicity of is a necessary condition here if is continuous. Then it is shown that quasimonotonicity can be verified by considering only a few positive continuous linear functionals in the definition (for instance in by taking coordinate functionals).

     

On topological spaces of singular density and minimal weight

We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω₁ is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.     

G-primitive extensions of differential fields

A classification is given of the G-primitive extensions of a partial differential field of characteristic zero with an algebraically closed field of constants and a connected algebraic group G.Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 425–434, March, 1978.     

Skew fields of differential operators

We show in this note that two curves are defined up to birational equivalence by the skew fields of differential operators on these curves. The project was started when this author was visiting Wayne State University. This author is supported by an NSF grant.     

Adequate predimension inequalities in differential fields

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the j-function (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterovi? and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate.     

Convex topological algebras via linear vector fields and Cuntz algebras

A realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is in analogue to the classical Jordan—Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schrödinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.