17 research outputs found
On a question of Abraham Robinson's
In this note we give a negative answer to Abraham Robinson's question whether
a finitely generated extension of an undecidable field is always undecidable.
We construct 'natural' undecidable fields of transcendence degree 1 over Q all
of whose proper finite extensions are decidable. We also construct undecidable
algebraic extensions of Q that allow decidable finite extensions
A Diophantine definition of the constants in
It is an old problem in the area of Diophantine definability to determine
whether is Diophantine in . We provide a positive
answer conditional on two standard conjectures on elliptic surfaces
First-order definitions in function fields over anti-Mordellic fields
A field k is called anti-Mordellic if every smooth curve over k with a
k-point has infinitely many k-points. We prove that for a function field over
an anti-Mordellic field, the subfield of constants is defined by a certain
universal first order formula. Under additional hypotheses regarding
2-cohomological dimension we prove that algebraic dependence of an n-tuple of
elements in such a function field can be described by a first order formula,
for each n. We also give a result that lets one distinguish various classes of
fields using first order sentences.Comment: 12 page
Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0
Let be a one-variable function field over a field of constants of
characteristic 0. Let be a holomorphy subring of , not equal to . We
prove the following undecidability results for : If is recursive, then
Hilbert's Tenth Problem is undecidable in . In general, there exist
such that there is no algorithm to tell whether a
polynomial equation with coefficients in \Q(x_1,...,x_n) has solutions in
.Comment: This version contains minor revisions and will appear in Annales de l
Institut Fourie
Undecidability in function fields of positive characteristic
We prove that the first-order theory of any function field K of
characteristic p>2 is undecidable in the language of rings without parameters.
When K is a function field in one variable whose constant field is algebraic
over a finite field, we can also prove undecidability in characteristic 2. The
proof uses a result by Moret-Bailly about ranks of elliptic curves over
function fields.Comment: 12 pages; strengthened main theorem, proved undecidability in the
language of rings without parameter
Panorama of p-adic model theory
ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud
Denef’s work on the rationality of Poincaré series. / RÉSUMÉ. Nous donnons un aperçu des développements de la théorie des modèles\ud
des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud
par une revue de la bibliographie