A remark on Ax's theorem on solvability modulo primes.
Elimination theory for the ring of algebraic integers.
The Field of Reals with a Predicate for the Powers of Two.
Dense pairs of o-minimal structures
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.
Subjective polynomial maps, and a remark on the jacobian problem.
Extending Tamm's theorem
We extend a result of M. Tamm as follows: Let , be definable in the ordered field of real numbers augmented by all real analytic functions on compact boxes and all power functions . Then there exists such that for all , if is in a neighborhood of , then is real analytic in a neighborhood of .
Fields of surreal numbers and exponentiation
We show that Conway's field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case,...
Erratum to "Fields of surreal numbers and exponentiation" (Fund. Math. 167 (2001), 173-188)
Triangulation in o-minimal fields with standard part map
In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V → k be the corresponding standard part map. Under a mild assumption on (R,V) we show that a definable set X ⊆ Vⁿ admits a triangulation that induces a triangulation of its standard part st X ⊆ kⁿ.
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