To logicians, arithmetic is horribly complicated. 1) is undecidable. Moreover, there are sets defined by arbitrarily complicated formulae (in the sense of quantifier complexity, say) which cannot be expressed in terms of simpler formulae. On the contary, geometry is very regular. Theorem 2 (Tarski) Euclidean geometry, understood as the theory of the real… (More)
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)
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)
The notion of a D-ring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the Ax-Kochen-Ershov principle is proven for a theory of valued D-fields of residual characteristic zero. The model theory of differential and difference fields has been extensively studied (see for example [7, 3]) and… (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 present the details of a model theoretic proof of an analogue of the Manin-Mumford conjecture for semiabelian varieties in positive characteristic. As a by-product of the proof we reduce the general positive characteristic Mordell-Lang problem to a question about purely inseparable points on sub-varieties of semiabelian varieties.
We prove versions of the Mordell-Lang conjecture for semiabelian varieties defined over fields of positive characteristic.
Communication, i.e., moving data, between levels of a memory hierarchy or between parallel processors on a network, can greatly dominate the cost of computation, so algorithms that minimize communication can run much faster (and use less energy) than algorithms that do not. Motivated by this, attainable communication lower bounds were established in [12,… (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)