2019/10/27 by Nigel Pynn-Coates, Pynn-Coates, Nigel
Computer Science · Mathematics · #Advanced Topology and Set Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Logic (math.LO) #Logic, programming, and type systems #Mathematical and Theoretical Analysis
paper · pdf · doi:10.48550/arxiv.1910.12171
openalex publication_date 2019/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.