arXiv subjects
Publications and source records attributed to Lou van den Dries.
We define the universal exponential extension of an algebraically closed differential field and investigate its properties in the presence of a nice valuation and in connection with linear differential equations. Next we prove normalization theorems for algebraic differential equations over $H$-fields, as a tool in solving such equations in suitable extensions. The results in this monograph are essential in our work on Hardy fields in [6].
We show that every Hardy field extends to an $ω$-free Hardy field. This result relates to classical oscillation criteria for second-order homogeneous linear differential equations. It is essential in [10], and here we apply it to answer questions of Boshernitzan, and to generalize a theorem of his.
We review second-order homogeneous linear differential equations with coefficient functions whose germs lie in a Hardy field (and hence are strongly non-oscillating). We prove a conjecture of Boshernitzan (1982): the oscillating solutions to such an equation are given by amplitude and phase functions with germs in a bigger Hardy field, and hence oscillate in a very regular way. We give sharp conditions for the uniqueness of such germs, study their asymptotic behavior, and use this to obtain information about the zeros and critical points of oscillating solutions.
We characterize intrinsically the truncation structures on valued fields arising from embeddings into Hahn fields with truncation closed image.
Let $C\subseteq M$ be stably embedded in a structure $\cM=(M;\dots)$. We consider {\em Fubini measures} on the subcategory $\Def(C)$ of the category $\Def(\cM)$ of definable sets in $\cM$, with ``Fubini" signaling good behaviour in definable families. We show that such a Fubini measure extends uniquely to the larger subcategory of $\Def(\cM)$ whose objects are the sets that are ``fiberable over $C$". In cases of interest ``fiberable over $C$" coincides with ``co-analyzable relative to $C$." This applies in particular to the differential field $\T$ of transseries with $C=\R$, and to differentially closed fields with constant field $C$.
We show that maximal analytic Hardy fields are $η_1$ in the sense of Hausdorff. We also prove various embedding theorems about analytic Hardy fields. For example, the ordered differential field $\mathbb T$ of transseries is shown to be isomorphic to an analytic Hardy field.
Differentially algebraic Hardy field extensions of short Hardy fields are short. This is proved in the more general setting of $H$-fields. As an application we extend a theorem of Rosenlicht (1981) by showing that each short asymptotic couple of Hardy type with small derivation is isomorphic to the asymptotic couple of an analytic Hardy field.
We show that all maximal Hardy fields are elementarily equivalent as differential fields, and give various applications of this result and its proof. We also answer some questions on Hardy fields posed by Boshernitzan.
We study relative differential closure in the context of Hardy fields. Using our earlier work on algebraic differential equations over Hardy fields, this leads to a proof of a conjecture of Boshernitzan (1981): the intersection of all maximal analytic Hardy fields agrees with that of all maximal Hardy fields. We also generalize a key ingredient in the proof, and describe a cautionary example delineating the boundaries of its applicability.
We develop here the algebra of the differential field of transseries and of related valued differential fields. This book contains in particular our recently obtained decisive positive results on the model theory of these structures.
We show that all maximal Hardy fields are elementarily equivalent as differential fields to the differential field $\mathbb T$ of transseries, and give various applications of this result and its proof.
We show how to fill "countable" gaps in Hardy fields. We use this to prove that any two maximal Hardy fields are back-and-forth equivalent.
We develop an extension theory for analytic valuation rings in order to establish Ax-Kochen-Ersov type results for these structures. New is that we can add in salient cases lifts of the residue field and the value group and show that the induced structure on the lifted residue field is just its field structure, and on the lifted value group is just its ordered abelian group structure. This restores an analogy with the non-analytic AKE-setting that was missing in earlier treatments of analytic AKE-theory.
Every discrete definable subset of a closed asymptotic couple with ordered scalar field $\boldsymbol k$ is shown to be contained in a finite-dimensional $\boldsymbol k$-linear subspace of that couple. It follows that the differential-valued field $\mathbb T$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb R$ as the underlying set for the second sort.
This expository paper gives an account of the Pila-Wilkie counting theorem and some of its extensions and generalizations. We use semialgebraic cell decomposition to simplify part of the original proof. We also include complete treatments of a result due to Pila and Bombieri and of the o-minimal Yomdin-Gromov theorem that are used in this proof. For the latter we follow Binyamini and Novikov.
Liouville closed $H$-fields are ordered differential fields whose ordering and derivation interact in a natural way and where every linear differential equation of order $1$ has a nontrivial solution. (The introduction gives a precise definition.) For a Liouville closed $H$-field $K$ with small derivation we show: $K$ has the Intermediate Value Property for differential polynomials iff $K$ is elementarily equivalent to the ordered differential field of transseries. We also indicate how this applies to Hardy fields.
So far there exist just a few results about the uniqueness of maximal immediate valued differential field extensions and about the relationship between differential-algebraic maximality and differential-henselianity; see arXiv:1509.02588, Chapter 7. We remove here the assumption of monotonicity in these results but replace it with the assumption that the value group is the union of its convex subgroups of finite (archimedean) rank. We also show the existence and uniqueness of differential-henselizations of asymptotic fields with such a value group.
We discuss the conjecture that every maximal Hardy field has the Intermediate Value Property for differential polynomials, and its equivalence to the statement that all maximal Hardy field are elementarily equivalent to the differential field of transseries. As a modest but essential step towards establishing the conjecture we show that every maximal Hardy field is $ω$-free.