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 . 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 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 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.
In this thesis we introduce a general notion of a D-ring generalizing that of a differential or difference ring. In Chapter 3, this notion is specialized to consider valued fields D-fields: valued fields K having an operator D : K → K and a fixed element e ∈ K satisfying D(x + y) = Dx + Dy, D(1) = 0, D(xy) = xDy + yDx + eDxDy, v(e) ≥ 0, and v(Dx) ≥ v(x).… (More)
Generalising and unifying the known theorems for difference and differential fields, it is shown that for every finite free S-algebra D over a field A of characteristic zero the theory of D-fields has a model companion D-CF 0 which is simple and satisfies the Zilber dichotomy for finite-dimensional minimal types.
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 introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety is determined by its arcs at a point. is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K, +) of the… (More)
Building on the abstract notion of prolongation developed in , the theory of iterative Hasse-Schmidt rings and schemes is introduced, simultaneously generalising difference and (Hasse-Schmidt) differential rings and schemes. This work provides a unified formalism for studying difference and differential algebraic geometry, as well as other related… (More)