arXiv subjects

Leo Gitin

Publications and source records attributed to Leo Gitin.

3 recordsLinked to original sources

arXiv2026

The complexity of tilting

Many explicit constructions in algebra are not first-order definable in classical finitary logic, but are, however, definable in infinitary logic. The aim of this paper is to show how infinitary logic can be used to lift model-theoretic results in arithmetic to a quantifier-sensitive form. We apply this idea to a deep result of Jahnke and Kartas on the model theory of perfectoid fields. Our main result is that a first-order sentence $ϕ$ holds in the tilt $k^{\flat}$ of a perfectoid field $k$ if and only if a certain sentence $ϕ^{\natural}$ holds in a non-principal ultrapower of $k$, where $ϕ^{\natural}$ does not depend on $k$ and is only slightly more complicated than $ϕ$ itself. In particular, tilting preserves the Lévy hierarchy in a precise sense. To this end, we prove that the Standard Decomposition of a valued field at a constant $t$ is infinitarily interpretable by formulas of rank at most 1.

math.LO↗

arXiv2026

The Galois characterisation of $p$-adically closed fields -- A modern perspective

In 1927, Artin and Schreier showed that a field is real closed if and only if its absolute Galois group has order two. Inspired by this characterisation and drawing on earlier work of Neukirch, Pop conjectured the following $p$-adic analogue: a field is $p$-adically closed if and only if its absolute Galois group is isomorphic to that of $\mathbb{Q}_p$. In 1995, the conjecture was independently solved by Efrat for $p \ne 2$ and by Koenigsmann in full generality. Using novel techniques in the theory of valued fields developed over the last 25 years, we give a new, elementary, and self-contained proof of this theorem, with a Galois characterisation of henselianity at the heart of the proof and without relying on Galois cohomology. We further highlight connections to the recent work of Jahnke-Kartas on perfectoid fields and model-theoretic transfer techniques. We provide a systematic account of all of our methods to encourage further investigations.

math.NT↗

arXiv2024

Undecidability of expansions of Laurent series fields by cyclic discrete subgroups

In 1987, Pheidas showed that the field of Laurent series $\mathbb{F}_q((t))$ with a constant for the indeterminate $t$ and a predicate for the natural powers $\{t^n \mid n > 0\}$ of $t$ is existentially undecidable. We show that the same result holds true if $t$ is replaced by any element $α$ of positive $t$-adic valuation.

math.LO↗