David Marker

Weakly o-minimal structures and real closed fields

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

Logarithmic-Exponential Power Series

We use generalized power series to construct algebraically a nonstandard model of the theory of the real field with exponentiation. This model enables us to show the undefinability of the zeta function and certain non-elementary and improper integrals. We also use this model to answer a question of Hardy by showing that the compositional inverse to the function (log x ) (log log x ) is not asymptotic as x →+∞ to a composition of semialgebraic functions, log and exp.

Logarithmic-exponential series

Abstract We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” (LE-series), which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered.

Non Sigma n Axiomatizable Almost Strongly Minimal Theories.

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ 0 -categorical almost strongly minimal theory is Σ n axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ 0 -categorical and almost strongly minimal, then T is Σ n axiomatizable for some n . This result also follows from Ahlbrandt and Zieglers results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyres conjecture by showing that for each n there is an ℵ 0 -categorical almost strongly minimal theory which is not Σ n axiomatizable. Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set. We will prove the following result. Theorem. For every n there is an almost strongly minimal ℵ 0 -categorical theory T with models M and N such that N is Σ n elementary but not Σ n + 1 elementary . To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C]. Theorem. If T is a Σ n axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σ n elementary extension of M, then N is an elementary extension of M .

Semialgebraic Expansions of C

We prove no nontrivial expansion of the field of complex numbers can be obtained from a reduct of the field of real numbers

The Borel Complexity of Isomorphism for Theories with Many Types

During the Notre Dame workshop on Vaught’s Conjecture, Hjorth and Kechris asked which Borel equivalence relations can arise as the isomorphism relation for models of a first order theory. In particular, they asked if the isomorphism relation can be essentially countable but not tame. We show this is not possible if the theory has uncountably many types. I am grateful to the logic group at Notre Dame for organizing this stimulating workhop.

Reducts of (C, +, ·) which contain +

We show that the structure (C, +, •) has no proper non locally modular reducts which contain +. In other words, if X⊂C n is constructible and not definable in the module structure (C, +, λ a ) a∈C (where λ a denotes multiplication by a) then multiplication is definable in (C, +, X)

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.

Khovanskii’s Theorem

Let f 1,...,f s : R n → R be C 1. We say that (f 1,...,f s ) is a Pfaffian chain if and only if