Lou van den Dries

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