Combinatorics with definable sets: Euler characteristics and Grothendieck rings

We recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counterexamples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

[1]  Patrick Suppes,et al.  Logic, Methodology and Philosophy of Science , 1963 .

[2]  Angus Macintyre.  Residue Fields of Models of P , 1982 .

[3]  Jeff B. Paris,et al.  Provability of the Pigeonhole Principle and the Existence of Infinitely Many Primes , 1988, J. Symb. Log..

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  Saharon Shelah,et al.  Every two elementarily equivalent models have isomorphic ultrapowers , 1971 .

[6]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[7]  Stephen H. Schanuel,et al.  Negative sets have Euler characteristic and dimension , 1991 .

[8]  J. Denef,et al.  Germs of arcs on singular algebraic varieties and motivic integration , 1999 .

[9]  Angus Macintyre,et al.  Non-standard finite fields overIΔ0+Ω1 , 2000 .

[10]  Horst Herrlich,et al.  Category theory , 1979 .

[11]  Jan Krajícek,et al.  Bounded arithmetic, propositional logic, and complexity theory , 1995, Encyclopedia of mathematics and its applications.

[12]  Jan Krajícek,et al.  Lower bounds on Hilbert's Nullstellensatz and propositional proofs , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[13]  L. van den Dries,et al.  Tame Topology and O-minimal Structures , 1998 .

[14]  Anand Pillay,et al.  Model theory of fields , 1996 .

[15]  Misha Gromov,et al.  Endomorphisms of symbolic algebraic varieties , 1999 .

[16]  Rohit Parikh,et al.  Existence and feasibility in arithmetic , 1971, Journal of Symbolic Logic.

[17]  Jan Krajícek,et al.  An Exponenetioal Lower Bound to the Size of Bounded Depth Frege Proofs of the Pigeonhole Principle , 1995, Random Struct. Algorithms.

[18]  F. Torres,et al.  Algebraic Curves over Finite Fields , 1991 .

[19]  H. Keisler,et al.  Handbook of mathematical logic , 1977 .

[20]  Anand Pillay,et al.  Model theory of algebraically closed fields , 1998 .

[21]  Russell Impagliazzo,et al.  Exponential lower bounds for the pigeonhole principle , 1992, STOC '92.

[22]  Jan Krajíček,et al.  Uniform Families of Polynomial Equations Over a Finite Field and Structures Admitting an Euler Characteristic of Definable Sets , 2000 .

[23]  A. Wilkie.  TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248) By L OU VAN DEN D RIES : 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), ISBN 0 521 59838 9 (Cambridge University Press, 1998). , 2000 .

[24]  J. Paris.  A Mathematical Incompleteness in Peano Arithmetic , 1977 .

[25]  JAN KRAJIÂCÏ,et al.  UNIFORM FAMILIES OF POLYNOMIAL EQUATIONS OVER A FINITE FIELD AND STRUCTURES ADMITTING AN EULER CHARACTERISTIC OF DEFINABLE SETS , 2000 .

[26]  Seinosuke Toda.  On the computational power of PP and (+)P , 1989, 30th Annual Symposium on Foundations of Computer Science.