Related papers: Representing Scott sets in algebraic settings

We take two approaches to classifying the complexity of Presburger models: Scott analysis and degree spectra. In particular, we investigate the possible Scott sentence complexities and possible degree spectra of models of Presburger…

We introduce the notion of saturated sets of primes of an algebraic number field and prove an analogue of Riemann's existence theorem for the decomposition groups of infinite stably saturated sets of primes.

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an…

Let G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove…

We study the structure of an algebraically closed field with extra function resembling the classical exponentiation on complex numbers.

A definable set in a pair (K, k) of algebraically closed fields is co-analyzable relative to the subfield k of the pair if and only if it is almost internal to k. To prove this and some related results for tame pairs of real closed fields…

I show that assuming PFA, every proper Scott set is the standard system of a model of PA. A Scott set X is proper if it is arithmetically closed and the quotient Boolean algebra X/Fin is a proper partial order.

We consider the expansion of the real field by the group of rational points of an elliptic curve over the rational numbers. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets…

We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory $\mathsf{VTC^0}$ are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically…

We show that any stack $\mathfrak{X}$ of finite type over a Noetherian scheme has a presentation $X \rightarrow \mathfrak{X}$ by a scheme of finite type such that $X(F) \rightarrow \mathfrak{X}(F)$ is onto, for every finite or real closed…

Given a finite set $S$ of places of a number field, we prove that the field of totally $S$-adic algebraic numbers is not Hilbertian.

We define a class of pre-ordered abelian groups that we call finite-by-Presburger groups, and prove that their theory is model-complete. We show that certain quotients of the multiplicative group of a local field of characteristic zero are…

We give necessary and sufficient conditions for a polynomially bounded o-minimal expansion of a real closed field (in a language of arbitrary cardinality) to be $\aleph_{\alpha}$-saturated. The conditions are in terms of the value group,…

Let $K$ be a number field, and $S$ a finite set of places in $K$ containing all infinite places. We present an implementation for solving the $S$-unit equation $x + y = 1$, $x,y \in\mathscr{O}_{K,S}^\times$ in the computer algebra package…

We give an exposition of results of Baldwin-Shelah on saturated free algebras, at the level of generality of complete first order theories $T$ with a saturated model $M$ which is in the algebraic closure of an indiscernible set. We then…

Building over recent results, we expand the basic theory of algebraic extensions to the realm of superfields -a field with multivalued sum and product-, showing that every superfield has a (unique up to isomorphism) strong algebraic…

We consider the intersection map on the family of non-empty $\omega$-Scott-open sets of the lattice of opens of a topological space. We prove that in a certain class of topological spaces the intersection map forms a continuous retraction…

We prove that the theories of fields with Hasse-Schmidt derivations corresponding to actions of formal groups admit model companions. We also give geometric axiomatizations of these model companions.

We show that the Schubert calculus of enumerative geometry is real, for special Schubert conditions. That is, for any such enumerative problem, there exist real conditions for which all the a priori complex solutions are real.

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…