Lou van den Dries’s research while affiliated with University of Illinois Urbana-Champaign and other places

What is this page?


This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our legitimate goal of comprehensive and accurate scientific recordkeeping. If you are this author and want this page removed, please let us know.

Publications (134)


Relative differential closure in Hardy fields
  • Preprint
  • File available

December 2024

·

6 Reads

Matthias Aschenbrenner

·

Lou van den Dries

·

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.

Download

Sketch of ρ.
Sketch of ϕ, θ, ζ in Lemma 2.3.
Constructing ϕ in the proof of Lemma 2.7.
The functions γ, δ, θ.
The bump function α.

+5

Filling gaps in Hardy fields

August 2024

·

19 Reads

·

2 Citations

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.



Analytic Ax-Kochen-Ersov theory with lifts of the residue field and value group

July 2024

·

2 Reads

·

1 Citation

Transactions of the American Mathematical Society Series B

We develop an extension theory for analytic valuation rings in order to establish Ax-Kochen-Ersov (AKE) 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 nonanalytic AKE-setting that was missing in earlier treatments of analytic AKE-theory.




Revisiting closed asymptotic couples

June 2022

·

31 Reads

·

2 Citations

Proceedings of the Edinburgh Mathematical Society

Every discrete definable subset of a closed asymptotic couple with ordered scalar field k{\boldsymbol {k}} is shown to be contained in a finite-dimensional k{\boldsymbol {k}} -linear subspace of that couple. It follows that the differential-valued field T\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 R\mathbb {R} as the underlying set for the second sort.




On the Pila-Wilkie theorem

March 2022

·

65 Reads

·

2 Citations

Expositiones Mathematicae

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.


Citations (58)


... By [ADH] and [4,9], they share the same first-order theory, as ordered differential fields. In [10] we show that under Cantor's Continuum Hypothesis (CH), any maximal Hardy field is in fact isomorphic to the ordered differential subfield No(ω 1 ) of No (consisting of the surreal numbers of countable length). In [2] this is also shown for maximal smooth and maximal analytic Hardy fields 9 Without even assuming CH, one can embed T into each maximal analytic Hardy field and also into No(ω 1 ), cf. ...

Reference:

Relative differential closure in Hardy fields
Filling gaps in Hardy fields

... In particular, it is a model of the complete theory T H . Thus maximal Hardy fields have the intermediate value property for differential polynomials as well, and this amounts to Theorem A, obtained here as a byproduct of more fundamental results. (A detailed account of the differential intermediate value property for H-fields is in [5].) We sketch the proof of our main result (Theorem 11.19) later in this introduction, after describing further consequences. ...

On a differential intermediate value property
  • Citing Article
  • March 2022

Revista de la Unión Matemática Argentina

... Working with transseries has been the centre of quite vibrant research including a recent result linking this to O-minimality, see [10], possibly offering a road forward to the 16th problem of Hilbert once the roadblock of understanding these proofs have passed. It is also worth noting that transseries have been used in contexts more explicitly related to the problem of Dulac, for example in [11]. ...

Asymptotic Differential Algebra and Model Theory of Transseries: (AMS-195)
  • Citing Book
  • June 2017

... The differential field H ⊇ T of hyperseries was constructed as a field in [BHK21], building on [DHK19], and as an elementary differential field extension of T in [Bag22]. Just to highlight a couple of points, H is a proper class that contains formally transexponential elements such as exp ω (x), which in H is a solution to the functional equation E ω (x + 1) = exp E ω (x) that cannot be solved in T. Similarly, Conway's field No of surreal numbers from [Con76] (see also [Gon86]) is a proper class containing every ordinal number. ...

Logarithmic hyperseries

Transactions of the American Mathematical Society

... Nevertheless, following Conway, in our informal remarks we freely refer to certain inductive definitions as "genetic".2 By contrast, work of Berarducci and Mantova[13,14], Aschenbrenner, van den Dries and van der Hoeven[5], Bagayoko[7], Bagayoko, van der Hoeven and Kaplan[10], Bagayoko and van der Hoeven[8,9] and others (e.g.[11,28,29,69]) has made significant progress toward viewing the surreals as an ordered differential field. This work aims to bring a robust theory of asymptotic differential algebra to all of No. Unlike the present work, which is concerned with derivations on surreal functions, the former work is concerned with derivations on surreal numbers. ...

Homogeneous universal H-fields
  • Citing Article
  • October 2018

Proceedings of the American Mathematical Society