Lou van den Dries

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The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays a special role in this description. Introduction. In a classical paper [8] A. Robinson proved the completeness of the theory of real closed fields with a(More)
Theorems of Ax, Kochen and Ersov tell us that the elementary theory of a henselian valuation ring of equal characteristic 0 is completely determined by the elementary theories of its value group and residue eld, see 1], 3], 4], and the references therein. This elementary classiication goes through even when a predicate is added for a eld of representatives(More)
Much more is known about cε(a, b), but this simple-to-prove upper bound suggests the proper formulation of the Euclidean’s (worst case) optimality among its peers—algorithms from rem: Conjecture. If an algorithm α computes gcd(x, y) from rem with time complexity cα(x, y), then there is a rational number r > 0 such that for infinitely many pairs a > b > 1,(More)