Lou van den Dries
Professor
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street (MC-382)
Urbana, Illinois 61801-2975
Office: vddries@math.uiuc.edu
Area of Specialization:
Model theory and its applications.
Lecture Notes
Preprints:
- (with P. Speissegger) O-minimal preparation theorems,
to appear in Quaderni di Matematica. [dvi file
or pdf file ]
- (with M. Aschenbrenner) Liouville closed H-fields, J. Pure Appl. Algebra 197 (2005) 83-139.
[dvi file or
pdf file ]
Abstract: H-fields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending R and fields of transseries over R are H-fields.) We prove basic facts about the location of zeros of differential
polynomials in Liouville closed H-fields, and study various constructions in the category of H-fields: closure
under powers, constant field extension, completion, and building H-fields with prescribed constant field and H-couple.
We indicate difficulties in obtaining a good model theory of H-fields, including an undecidability result.
We finish with open questions that motivate our work.
- Generating the Greatest Common
Divisor, and Limitations of Primitive Recursive Algorithms, Found. Comput. Math. 3 (2003) 297-3224.
[ps file or
pdf file ]
Abstract:
The greatest common divisor of two integers cannot be generated in a uniformly bounded number of steps from those
integers using arithmetic operations. The proof uses an elementary model-theoretic construction that enables us to
focus on "integers with transcendental ratio". This unboundedness result is
part of the solution of a problem posed by Y. Moschovakis on limitations of primitive recursive algorithms
for computing the greatest common divisor function.
- (with A.J. Wilkie) The Laws of Integer Divisibility and Solution Sets of Linear Divisibility Conditions,
J. Symbolic Logic 68 (2003) 503-526.
[ps file or
pdf file ]
Abstract: We prove linear and polynomial growth properties of sets and functions that are existentially
definable in the ordered group of integers with divisibility. We determine the laws of addition with order
and divisibility.
Last modified January 11, 2007