# Lothar Sebastian KrappUniversity of Zurich | UZH

Lothar Sebastian Krapp

Dr.rer.nat.

## About

22

Publications

445

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50

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Introduction

Mathematical logic, model theory, algebraic aspects of ordered structures, non-archimedean fields

**Skills and Expertise**

## Publications

Publications (22)

The Fundamental Theorem of Statistical Learning states that a hypothesis space is PAC learnable if and only if its VC dimension is finite. For the agnostic model of PAC learning, the literature so far presents proofs of this theorem that often tacitly impose several measurability assumptions on the involved sets and functions. We scrutinize these p...

We give an explicit algebraic characterisation of all definable henselian valuations on a dp-minimal real field. Additionally we characterise all dp-minimal real fields that admit a definable henselian valuation with real closed residue field. We do so by first proving this for the more general setting of almost real closed fields.

Motivated by the decidability question for the theory of real exponentiation and by the Transfer Conjecture for o‐minimal exponential fields, we show that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation is embeddable into any o‐minimal exponential field, where the embedding is not necessarily elementary. This...

We develop a first-order theory of ordered transexponential fields in the language $\{+,\cdot,0,1,<,e,T\}$, where $e$ and $T$ stand for unary function symbols. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and...

Motivated by the decidability question for the theory of real exponentiation and by the Transfer Conjecture for o-minimal exponential fields, we show that, under the assumption of Schanuel's Conjecture, the prime model of real exponentiation is embeddable into any o-minimal exponential field. This is deduced from a more general unconditional result...

We study the definability of convex valuations on ordered fields, with a particular focus on the distinguished subclass of henselian valuations. In the setting of ordered fields, one can consider definability both in the language of rings $\mathcal{L}_{\mathrm{r}}$ and in the richer language of ordered rings $\mathcal{L}_{\mathrm{or}}$. We analyse...

Within the field of Laurent series with rational coefficients, those series representing rational functions can be characterised via linear recurrence relations of their coefficients. Inspired by this result, we establish a notion of generalised linear recurrence relations for fields of generalised power series. We study distinguished Hahn fields t...

Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show th...

An exponential $\exp $ on an ordered field $(K,+,-,\cdot ,0,1, is an order-preserving isomorphism from the ordered additive group $(K,+,0, to the ordered multiplicative group of positive elements $(K^{>0},\cdot ,1, . The structure $(K,+,-,\cdot ,0,1, is then called an ordered exponential field (cf. [6]). A linearly ordered structure $(M, is called...

In this note, we study substructures of generalized power series fields induced by families of well-ordered subsets of the group of exponents. We characterize the set-theoretic and algebraic properties of the induced substructures in terms of conditions on the families. We extend the work of Rayner by giving both necessary and sufficient conditions...

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of stron...

Given a henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show th...

In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to o...

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each mo...

The following conjecture is due to Shelah-Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of stron...

In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah-Hasson Conjecture (specialised to or...

In this note, we study substructures of generalised power series fields induced by families of well-ordered subsets of the group of exponents. We relate set theoretic and algebraic properties of the families to algebraic features of the induced sets. By this, we extend the work of Rayner ('An algebraically closed field', 1968) to truncation closed...

An exponential exp on an ordered field (K, +, −, ·, 0, 1, <) is an order-preserving isomorphism from the ordered additive group (K, +, 0, <) to the ordered multiplicative group of positive elements(K^(>0), ·, 1, <). The structure (K, +, −, ·, 0, 1, <, exp) is then called an ordered exponential field. A linearly ordered structure (M, <, ...) is call...

Let F be an Archimedean field, G a divisible ordered abelian group and h a group exponential on G. A triple (F, G, h) is realised in a non-Archimedean exponential field (K, exp) if the residue field of K under the natural valuation is F and the induced exponential group of (K, exp) is (G, h). We give a full characterisation of all triples (F, G, h)...

The following conjecture is due to Shelah$-$Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or it admits a non-trivial definable henselian valuation, in the language $\mathcal{L}_{\mathrm{r}} = \{+,-,\cdot,0,1\}$. Inspired by this, we formulate an analogous conjecture for ordered fields in the language $\mathcal...