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We study algebraic dynamical systems (and, more generally, σ-varieties) Φ : A n C → A n C given by coordinatewise univariate polynomials by refining an old theorem of Ritt on compositional identities amongst polynomials. More precisely, we find a nearly canonical way to write a polynomial as a composition of " clusters " from which one may easily read off(More)
We show that a connected group interpretable in a compact complex manifold (a meromorphic group) is definably an extension of a complex torus by a linear algebraic group, generalizing results in [4]. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal modular meromorphic group is a complex torus,(More)
The following propositions seem both plausible in their own right and apparently inconsistent: (1) Moral judgements like 'It is right that I V' ('valuations' for short) express beliefs; in this case, a belief about the rightness of my D-ing. (2) There is some sort of a necessary connection between being in the state thejudgement 'It is right that I '(More)
Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination and stability result for finitely generated modules over certain finite simple extensions of the integers given together with predicates for cycles of the distinguished generator of(More)
We give axiomatizations and quantifier eliminations for first-order theories of finitely ramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from(More)
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(More)
The notion of a prolongation of an algebraic variety is developed in an abstract setting that generalises the difference and (Hasse) differential contexts. An interpolating map that compares these prolongation spaces with algebraic jet spaces is introduced and studied.
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  • Stephen G Simpson, Frontmatter More, Michael Benedikt, Michael Glanzberg, Carl G Jockusch, Michael Rathjen +5 others
  • 2009
Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are(More)