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David Marker

Publications and source records attributed to David Marker.

10 recordsLinked to original sources
arXiv2026

Finding suitably generic points on curves with an application to the construction of rigid real closed fields

Let $K$ be an algebraically closed field of characteristic 0 and transcendence degree at least 2. Let $C\subset K^2$ be an irreducible curve defined over $K$ but not defined over the algebraic closure of $\mathbb Q$. There is $(x ,y)$ a $K$-point of $C$ such that $x$ and $y$ are algebraically independent. Moreover, if $C_0$ and $C_1$ are two such curves and there is a finite-to-finite algebraic correspondence between them defined over $K$, then there are corresponding $K$-points $(x_0,y_0)\in C_0$ and $(x_1,y_1)\in C_1$ such that $x_0$ and $y_0$ are algebraically independent and $x_1$ and $y_1$ are algebraically independent. We use the latter result to construct non-Archimedean real closed fields of transcendence degree $\kappa$ with no non-trivial automorphisms for all $2\le\kappa\le \aleph_1$.

math.LO↗
arXiv2024

Rigid Real Closed Fields

We construct a non-Archimedean real closed field of transcendence degree two with no non-trivial automorphisms

math.LO↗
arXiv2022

On the equations of Poizat and Li\'enard

We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Li\'enard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Li\'enard equations.

math.CA↗
arXiv2020

Anti-classification results for groups acting freely on the line

We explore countable ordered Archimedean groups from the point of view of descriptive set theory. We introduce the space of Archimedean left-orderings $\mathrm{Ar}(G)$ for a given countable group $G$, and prove that the equivalence relation induced by the natural action of $\mathrm{GL}_2(\mathbb{Q})$ on $\mathrm{Ar}(\mathbb{Q}^2)$ is not concretely classifiable. Then we analyze the isomorphism relation for countable ordered Archimedean groups, and pin its complexity in terms of the hierarchy of Hjorth, Kechris and Louveau. In particular, we show that its potential class is not $\boldsymbol{\Pi}^0_3$. This topological constraint prevents classifying Archimedean groups using countable subsets of reals. We obtain analogous results for the bi-embeddability relation, and we consider similar problems for circularly ordered groups, and o-minimal structures such as ordered divisible Abelian groups, and real closed fields. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice.

math.LO↗
arXiv2017

Scattered Sentences have Few Separable Randomizations

In the paper "Randomizations of Scattered Sentences", Keisler showed that if Martin's axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is "yes". It follows that the absolute Vaught conjecture holds if and only if every $L_{\omega_1\omega}$-sentence with few separable randomizations has countably many countable models.

math.LO↗
arXiv2014

Turing degree spectra of differentially closed fields

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 closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial countable graphs.

math.LO↗
arXiv2012

Uncountable Real Closed Fields with PA Integer Parts

D'Aquino, Knight and Starchenko classified the countable real closed fields with integer parts that are nonstandard models of Peano Arithmetic. We rule out some possibilities for extending their results to the uncountable and study real closures of $\omega_1$-like models of PA.

math.LO↗
arXiv2006

Decidability of the Natural Numbers with the Almost-All Quantifier

We consider the fragment F of first order arithmetic in which quantification is restricted to ''for all but finitely many.'' We show that the integers form an F-elementary substructure of the real numbers. Consequently, the F-theory of arithmetic is decidable.

math.LO↗