A differential ring is a ring equipped with a derivation. In this paper, we introduce integral rings. An integral ring is a ring equipped with a derivation and an integration. We study the basic properties of integral rings.
Published in Algebra universalis
We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a +\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepa...
In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of any Fuchsian group of the first kind and any genus. Our techniques combine tools from differential geometry, differential algebra and the model theory ...
Published in Algebra universalis
New examples of existentially closed Abelian lattice-ordered groups, possessing most of the known properties of infinitely generic Abelian lattice-ordered groups, are constructed using Fraïssé limits.
Abstract: We continue the study ofn-dependent groups, fields and related structures, largely motivated by the conjecture that everyn-dependent field is dependent. We provide evidence toward this conjecture by showing that every infiniten-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Con...
Published in Algebra universalis
The class of identical inclusions was defined by Lyapin. We prove that any set of identical inclusions in the class of semilattices is equivalent to an elementary (the first order) formula. Elementary identical inclusions forms the class of universal formulas which is situated strictly between identities and universal positive formulas. We describe...
AbstractWe prove that every ultraproduct of p-adics is inp-minimal (i.e., of burden 1). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.
We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.
Published in Foundations of Computational Mathematics
Let K be an effective field of characteristic zero. An effective tribe is a subset of K[[z1,z2,…]]=K∪K[[z1]]∪K[[z1,z2]]∪⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$...
Mekler's construction gives an interpretation of any structure in a finite relational language in a group (nilpotent of class $2$ and exponent $p>2$, but not finitely generated in general). Even though this construction is not a bi-interpretation, it is known to preserve some model-theoretic tameness properties of the original structure including s...
