15 research outputs found
The existential theory of equicharacteristic henselian valued fields
We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of Fq((t))
Existentially generated subfields of large fields
We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated.Let L be a large field of characteristic exponent p, and let E \subseteq L be an infinite existentially
generated subfield. We show that E contains L^(p^n), the p^n-th powers in L, for some n < ω.
This generalises a result of Fehm, which shows E = L, under the assumption that L is perfect. Our method is to first study existentially generated subfields of henselian fields. Since L is existentially closed in the henselian field L((t)), our result follows
Axiomatizing the existential theory of Fq((t))
We study the existential theory of equicharacteristic henselian valued fields
with a distinguished uniformizer. In particular, assuming a weak consequence of
resolution of singularities, we obtain an axiomatization of - and therefore an
algorithm to decide - the existential theory relative to the existential theory
of the residue field. This is both more general and works under weaker
resolution hypotheses than the algorithm of Denef and Schoutens, which we also
discuss in detail. In fact, the consequence of resolution of singularities our
results are conditional on is the weakest under which they hold true.Comment: New Remark 4.18 and expanded Remark 2.
Denseness results in the theory of algebraic fields
We study when the property that a field is dense in its real and p-adic
closures is elementary in the language of rings and deduce that all models of
the theory of algebraic fields have this property.Comment: 19 page
A survey of local-global methods for Hilbert's Tenth Problem
Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide
correctly, for each , whether the
diophantine equation has a solution in R. The celebrated
`Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for
is unsolvable, i.e.~there is no such algorithm. Since then,
Hilbert's Tenth Problem has been studied in a wide range of rings and fields.
Most importantly, for {number fields and in particular for }, H10
is still an unsolved problem. Recent work of Eisentr\"ager, Poonen,
Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward
what is known in this area, and has made essential use of local-global
principles for quadratic forms, and for central simple algebras. We give a
concise survey and introduction to this particular rich area of interaction
between logic and number theory, without assuming a detailed background of
either subject. We also sketch two further directions of future research, one
inspired by model theory and one by arithmetic geometry
A survey of local-global methods for Hilbert's Tenth Problem
Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each , whether the diophantine equation has a solution in R. The celebrated `Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for is unsolvable, i.e.~there is no such algorithm. Since then, Hilbert's Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for {number fields and in particular for }, H10 is still an unsolved problem. Recent work of Eisentr\"ager, Poonen, Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward what is known in this area, and has made essential use of local-global principles for quadratic forms, and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry
Characterizing Diophantine Henselian valuation rings and valuation ideals
We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal