Model theory of fields: An application to positive semidefinite polynomials
Nice local global fields. IV.
Notes on exponential-logarithmic terms
O-minimal fields with standard part map
Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V → k. Let be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in and conditions on (R,V) which imply o-minimality of . We also show that if R is ω-saturated and V is the convex hull of ℚ in R, then the sets definable in are exactly the standard parts of the sets definable in (R,V).
o-minimality.
On a universal axiomatization of the real closed fields
This paper presents a natural axiomatization of the real closed fields. It is universal and admits quantifier elimination.
On elementary equivalence, isomorphism and isogeny
Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...
On Ø-definable elements in a field
We develop an arithmetic characterization of elements in a field which are first-order definable by a parameter-free existential formula in the language of rings. As applications we show that in fields containing any algebraically closed field only the elements of the prime field are existentially ∅-definable. On the other hand, many finitely generated extensins of Q contain existentially ∅-definable elements which are transcendental over Q. Finally, we show that all transcendental elements in...
On places of algebraic function fields.
On the Galois Theory of function fields of one variable over number fields.
On the model companion for e-fold p-adically valued fields.
On valued function fields II. Regular functions and elements with the uniqueness property.
p-adic semi-algebraic sets and cell decomposition.
p-Henselian fields.
Properties of extensions of algebraically maximal fields.
We prove some properties similar to the theorem Ax-Kochen-Ershov, in some cases of pairs of algebraically maximal fields of residue characteristic p > 0. This properties hold in particular for pairs of Kaplansky fields of equal characteristic, formally p-adic fields and finitely ramified fields. From that we derive results about decidability of such extensions.
Real closed exponential fields
Ressayre considered real closed exponential fields and “exponential” integer parts, i.e., integer parts that respect the exponential function. In 1993, he outlined a proof that every real closed exponential field has an exponential integer part. In the present paper, we give a detailed account of Ressayre’s construction and then analyze the complexity. Ressayre’s construction is canonical once we fix the real closed exponential field R, a residue field section k, and a well ordering ≺ on R. The...
Remarks on existentially closed fields and diophantine equations
Schanuel Nullstellensatz for Zilber fields
We characterize the unsolvable exponential polynomials over the exponential fields introduced by Zilber, and deduce Picard's Little Theorem for such fields.
Some results about exponential fields (survey)
Sur la complexité du principe de Tarski-Seidenberg