Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

DOI10.2307/421058zbMath0968.03036OpenAlexW2033498708MaRDI QIDQ4527935

Jan Krajíček, Thomas J. Scanlon

Publication date: 5 July 2001

Published in: Bulletin of Symbolic Logic (Search for Journal in Brave)

Full work available at URL: http://www.math.ucla.edu/~asl/bsl/0603-toc.htm

zbMATH Keywords

Euler characteristicsbounded arithmeticGrothendieck ringcounting functionsfirst-order structurelocally finite structures

Mathematics Subject Classification ID

Complexity classes (hierarchies, relations among complexity classes, etc.) (68Q15) Models of arithmetic and set theory (03C62) Complexity of proofs (03F20) Basic properties of first-order languages and structures (03C07)

Related Items (21)

Counting mod \(n\) in pseudofinite fields ⋮ The Grothendieck ring of varieties and of the theory of algebraically closed fields ⋮ Grothendieck ring of the pairing function without cycles ⋮ O-MINIMALISM ⋮ Invariant measures in simple and in small theories ⋮ The surjectivity and the continuity of definable functions in some definably complete locally o-minimal expansions and the Grothendieck ring of almost o-minimal structures ⋮ On the Grothendieck ring of varieties ⋮ Grothendieck rings of Laurent series fields. ⋮ Poor triviality and the sameness of Grothendieck semirings ⋮ Weakly o-minimal nonvaluational structures ⋮ Grothendieck rings of theories of modules ⋮ Grothendieck Rings of ℤ-Valued Fields ⋮ One-dimensional asymptotic classes of finite structures ⋮ The Grothendieck ring of varieties and piecewise isomorphisms ⋮ Grothendieck rings of o-minimal expansions of ordered Abelian groups ⋮ On the Euler characteristic of definable groups ⋮ Euler characteristics for strongly minimal groups and the eq-expansions of vector spaces ⋮ Definable combinatorics with dense linear orders ⋮ A model-theoretic characterization of the weak pigeonhole principle ⋮ MULTIDIMENSIONAL EXACT CLASSES, SMOOTH APPROXIMATION AND BOUNDED 4-TYPES ⋮ On braided zeta functions

Cites Work

• Exponential lower bounds for the pigeonhole principle
• Germs of arcs on singular algebraic varieties and motivic integration
• Endomorphisms of symbolic algebraic varieties
• The elementary theory of finite fields
• Every two elementarily equivalent models have isomorphic ultrapowers
• Provability of the pigeonhole principle and the existence of infinitely many primes
• An exponential lower bound to the size of bounded depth frege proofs of the pigeonhole principle
• Existence and feasibility in arithmetic

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