Model theory of differential-henselian pre-$H$-fields
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2019
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914309084282880 |
|---|---|
| author | Pynn-Coates, Nigel |
| author_facet | Pynn-Coates, Nigel |
| contents | Pre-$H$-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-$H$-fields that are differential-Hensel-Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax--Kochen/Ershov theorem for such structures: the theory of a differential-Hensel-Liouville closed pre-$H$-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel-Liouville closed pre-$H$-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-$H$-field sort, from which we deduce that the ordered differential residue field is stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel-Liouville closed pre-$H$-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-$H$-fields with gap~$0$, and is complete, distal, and locally o-minimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_12171 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Model theory of differential-henselian pre-$H$-fields Pynn-Coates, Nigel Logic Commutative Algebra Pre-$H$-fields are ordered valued differential fields satisfying some basic axioms coming from transseries and Hardy fields. We study pre-$H$-fields that are differential-Hensel-Liouville closed, that is, differential-henselian, real closed, and closed under exponential integration, establishing an Ax--Kochen/Ershov theorem for such structures: the theory of a differential-Hensel-Liouville closed pre-$H$-field is determined by the theory of its ordered differential residue field; this result fails if the assumption of closure under exponential integration is dropped. In a two-sorted setting with one sort for a differential-Hensel-Liouville closed pre-$H$-field and one sort for its ordered differential residue field, we eliminate quantifiers from the pre-$H$-field sort, from which we deduce that the ordered differential residue field is stably embedded and if it has NIP, then so does the two-sorted structure. Similarly, the one-sorted theory of differential-Hensel-Liouville closed pre-$H$-fields with closed ordered differential residue field has quantifier elimination, is the model completion of the theory of pre-$H$-fields with gap~$0$, and is complete, distal, and locally o-minimal. |
| title | Model theory of differential-henselian pre-$H$-fields |
| topic | Logic Commutative Algebra |
| url | https://arxiv.org/abs/1910.12171 |