Dimension theory for the asymptotic couple of the field of logarithmic transseries

2025/07/02 by Gehret, Allen, Kaplan, Elliot, Pynn-Coates, Nigel
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2507.01760

Abstract

In this paper we completely characterize all dimension functions on all models of the theory T_log of the asymptotic couple of the field of logarithmic transseries (Dimension Theorem). This is done by characterizing the "small" 1-variable definable sets (Small Sets Theorem). As a byproduct, we show that T_log is d-minimal and does not eliminate imaginaries. Separately, we provide an abstract criterion for d-minimality, which we use to observe some new examples of d-minimal expansions of valued fields.

Citations

• Measurability in the Fundamental Theorem of Statistical Learning
• Tameness Properties in Multiplicative Valued Difference Fields with Lift and Section
• Visceral theories without assumptions
• Analytic Number Theory and Algebraic Asymptotic Analysis
• Revisiting closed asymptotic couples
• D-minimal structures
• Distality in valued fields and related structures
• Cutting lemma and Zarankiewicz's problem in distal structures
• Dimension in the realm of transseries
• Strong theories of ordered abelian groups
• Regularity lemma for distal structures
• NIP for the Asymptotic Couple of the Field of Logarithmic Transseries
• Additivity of the dp-rank
• Dp-minimality: basic facts and examples

Related

• The Asymptotic Couple of the Field of Logarithmic Transseries
• On existence of log minimal models II
• NIP for the Asymptotic Couple of the Field of Logarithmic Transseries
• Dimension in the realm of transseries
• The nef dimension of log minimal models