Dimension and topology in transserial tame pairs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909778036391936 |
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| author | Pynn-Coates, Nigel |
| author_facet | Pynn-Coates, Nigel |
| contents | Every maximal Hardy field has a proper elementary differential subfield that is Dedekind complete in the maximal Hardy field. This pair of Hardy fields is a transserial tame pair, shown to have a complete and model complete elementary theory in arXiv:2408.07033. This paper introduces a dimension in transserial tame pairs and shows that it equals the (naive) dimension coming from the order topology. In particular, the dimension is definable in a certain sense, and is the unique definable dimension in a transserial tame pair. Further, transserial tame pairs are locally o-minimal and d-minimal. These properties and the dimension are used to establish topological properties of definable sets in transserial tame pairs, including a definable Baire category theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_16415 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Dimension and topology in transserial tame pairs Pynn-Coates, Nigel Logic Commutative Algebra Every maximal Hardy field has a proper elementary differential subfield that is Dedekind complete in the maximal Hardy field. This pair of Hardy fields is a transserial tame pair, shown to have a complete and model complete elementary theory in arXiv:2408.07033. This paper introduces a dimension in transserial tame pairs and shows that it equals the (naive) dimension coming from the order topology. In particular, the dimension is definable in a certain sense, and is the unique definable dimension in a transserial tame pair. Further, transserial tame pairs are locally o-minimal and d-minimal. These properties and the dimension are used to establish topological properties of definable sets in transserial tame pairs, including a definable Baire category theorem. |
| title | Dimension and topology in transserial tame pairs |
| topic | Logic Commutative Algebra |
| url | https://arxiv.org/abs/2508.16415 |