arXiv subjects

Sylvy Anscombe

Publications and source records attributed to Sylvy Anscombe.

At least 19 recordsLinked to original sources

arXiv2026

Existential fragments of theories of henselian valued fields

We study fragments of the existential theory of henselian valued fields with parameters. This includes the $\exists_n$-fragment in the equicharacteristic or unramified mixed characteristic case, the $\exists_n\exists_1$-fragment in the equicharacteristic case, and the $\exists_n$-fragment in the residue characteristic zero case. For example, we obtain an unconditional axiomatization (and thereby decidability) of the $\exists_3$-theory of $\mathbb{F}_{q}(\!(t)\!)$ in the language of valued fields with a parameter for $t$.

math.LO↗

arXiv2026

Elimination results for tame fields with finite residue fields

Building on work of Kuhlmann and Lisinski, we study the theory of the Hahn series field $\mathbb{F}_{q}(\!(\mathbb{Q})\!)$, over a finite field $\mathbb{F}_{q}$, equipped with the $t$-adic valuation, in a language of valued fields. We prove that every formula is equivalent to a formula $\exists y\colon f(x_{1},\ldots,x_{n},y)=0$, for a polynomial $f\in\mathbb{Z}[x_{1},\ldots,x_{n},y]$.

math.LO↗

arXiv2026

A note on existentially t-henselian fields

A field is existentially t-henselian if it is has the same existential theory in the first-order language of rings as a field that admits a nontrivial henselian valuation. This property turns out to be equivalent to $\mathbb{Z}$-largeness, which is a property identified in previous work with Fehm, and which holds for $F$ if and only if $tF[\![t]\!]$ is not Diophantine in $F(\!(t)\!)$, without extra constants. In this short note, we further investigate this property in order to count the number of existential theories of henselian valuations on a given field, and to find other characterizations of existential t-henselianity.

math.LO↗

arXiv2026

The model theory of perfectoid fields [after Jahnke and Kartas]

This text was written to support a Bourbaki seminar given in January 2026 on the subject of the model theory of perfectoid fields, especially on the work of Jahnke and Kartas in their paper "Beyond the Fontaine-Wintenberger theorem", J. Amer. Math. Soc. 38 (4), pp. 997-1047, 2025.

math.LO↗

arXiv2026

Ax-Kochen-Ershov principles for finitely ramified henselian fields

We study the model theory of finitely ramified henselian valued fields of fixed initial ramification, obtaining versions of the Ax-Kochen-Ershov principle as follows. We identify the induced structure on the residue field and show that once the residue field is endowed with this structure, the theory of the valued field is determined by the theories of the enriched residue field and the value group. Similarly, we show that the existential theory of the valued field is determined by the positive existential theory of the enriched residue field. We also prove that an embedding of finitely ramified henselian valued fields is existentially closed as soon as the induced embeddings of value group and residue field are existentially closed. This last result requires no enrichment of the residue field, in analogy to the corresponding result for model completeness, which holds by results of Ershov and Ziegler.

math.LO↗

arXiv2026

Universal-existential theories of fields

We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

math.LO↗

arXiv2025

On Lambda functions in henselian and separably tame valued fields

Given a field extension $F/C$, the ``Lambda closure'' $Λ_{F}C$ of $C$ in $F$ is a subextension of $F/C$ that is minimal with respect to inclusion such that $F/Λ_{F}C$ is separable. The existence and uniqueness of $Λ_{F}C$ was proved by Deveney and Mordeson in 1977. We show that it admits a simple description in terms of given generators for $C$: we expand the language of rings by the parameterized Lambda functions, and then $Λ_{F}C$ is the subfield of $F$ generated over $C$ by additionally closing under these functions. We then show that, given particular generators of $C$, $Λ_{F}C$ is the subfield of $F$ generated iteratively by the images of the generators under Lambda functions taken with respect to $p$-independent tuples also drawn from those generators. We apply these results to given a ``local description'' of existentially definable sets in fields equipped with a henselian topology. Let $X(K)$ be an existentially definable set in the theory of a field $K$ equipped with a henselian topology $τ$. We show that there is a definable injection into $X(K)$ from a Zariski-open subset $U_{1}^{\circ}$ of a set with nonempty $τ$-interior, and that each element of $U_{1}^{\circ}$ is interalgebraic (over parameters) with its image in $X(K)$. This can be seen as a kind of {\em very weak local quantifier elimination}, and it shows that existentially definable sets are (at least generically and locally) definably pararameterized by ``big'' sets. In the final section we extend the theory of separably tame valued fields, developed by Kuhlmann and Pal, to include the case of infinite degree of imperfection, and to allow expansions of the residue field and value group structures. We prove an embedding theorem which allows us to deduce the usual kinds of resplendent Ax--Kochen/Ershov principles.

math.LO↗

arXiv2024

Interpretations of syntactic fragments of theories of fields

We set up general machinery to study interpretations of fragments of theories. We then apply this to existential fragments of theories of fields, and especially of henselian valued fields. As an application we prove many-one reductions between various existential theories of fields. In particular we exhibit several theories of fields many-one equivalent to the existential theory of $\mathbb{Q}$.

math.LO↗

arXiv2024

Multidimensional asymptotic classes

We develop a general framework (multidimensional asymptotic classes, or m.a.c.s) for handling classes of finite first order structures with a strong uniformity condition on cardinalities of definable sets: The condition asserts that definable families given by a formula ϕ(x,y) should take on a fixed number n_ϕof approximate sizes in any M in the class, with those sizes varying with M. The prototype is the class of all finite fields, where the uniformity is given by a theorem of Chatzidakis, van den Dries and Macintyre. It inspired the development of asymptotic classes of finite structures, which this new framework extends. The underlying theory of m.a.c.s is developed, including preservation under bi-interpretability, and a proof that for the m.a.c. condition to hold it suffices to consider formulas ϕ(x,y) with x a single variable. Many examples of m.a.c.s are given, including 2-sorted structures (F,V) where V is a vector space over a finite field F possibly equipped with a bilinear form, and an example arising from representations of quivers of finite representation type. We also give examples and structural results for multidimensional exact classes (m.e.c.s), where the definable sets take a fixed number of precisely specified cardinalities, which again vary with M. We also develop a notion of infinite generalised measurable structure, whereby definable sets are assigned values in an ordered semiring. We show that any infinite ultraproduct of a m.a.c. is generalised measurable, that values can be taken in an ordered ring if the m.a.c. is a m.e.c., and explore model-theoretic consequences of generalised measurability. Such a structure cannot have the strict order property, and stability-theoretic properties can be read off from the measures in the semiring.

math.LO↗

arXiv2024

One-dimensional F-definable sets in F((t))

In this note we study one-dimensional definable sets in power series fields with perfect residue fields. Using the description of automorphisms given by Schilling, in \cite{S44}, we show that such sets are unions of existentially definable in the language of rings, allowing parameters. We deduce that if $F$ is a perfect field of positive characteristic $p$, and $X$ is a subset of the $t$-adically valued $F((t))$ that is definable in the language of valued fields with parameters from $F$, then the subfield $(X)$ generated by $X$ is either contained in $F$ or equal to $F((t^{p^n}))$, for some $n\geq0$. The proof uses our earlier work on existentially definable subsets of henselian and large fields, of which power series fields are examples.

math.LO↗

arXiv2023

Characterizing NIP henselian fields

In this paper, we characterize NIP henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

math.LO↗

arXiv2023

A survey of local-global methods for Hilbert's Tenth Problem

Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide correctly, for each $f\in\mathbb{Z}[X_{1},\dots,X_{n}]$, whether the diophantine equation $f(X_{1},...,X_{n})=0$ has a solution in R. The celebrated `Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for $\mathbb{Z}$ is unsolvable, i.e.~there is no such algorithm. Since then, Hilbert's Tenth Problem has been studied in a wide range of rings and fields. Most importantly, for {number fields and in particular for $\mathbb{Q}$}, H10 is still an unsolved problem. Recent work of Eisenträger, Poonen, Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward what is known in this area, and has made essential use of local-global principles for quadratic forms, and for central simple algebras. We give a concise survey and introduction to this particular rich area of interaction between logic and number theory, without assuming a detailed background of either subject. We also sketch two further directions of future research, one inspired by model theory and one by arithmetic geometry.

math.NT↗

arXiv2023

Axiomatizing the existential theory of Fq((t))

We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of - and therefore an algorithm to decide - the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses than the algorithm of Denef and Schoutens, which we also discuss in detail. In fact, the consequence of resolution of singularities our results are conditional on is the weakest under which they hold true.

math.LO↗

arXiv2021

The model theory of Cohen rings

The aim of this article is to give a self-contained account of the algebra and model theory of Cohen rings, a natural generalization of Witt rings. Witt rings are only valuation rings in case the residue field is perfect, and Cohen rings arise as the Witt ring analogon over imperfect residue fields. Just as one studies truncated Witt rings to understand Witt rings, we study Cohen rings of positive characteristic as well as of characteristic zero. Our main results are a relative completeness and a relative model completeness result for Cohen rings, which imply the corresponding Ax-Kochen/Ershov type results for unramified henselian valued fields also in case the residue field is imperfect.

math.LO↗

arXiv2019

Denseness results in the theory of algebraic fields

We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.

math.LO↗

arXiv2019

Approximation theorems for spaces of localities

The classical Artin--Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.

math.AC↗

arXiv2019

A p-adic analogue of Siegel's Theorem on sums of squares

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.

math.NT↗

arXiv2017

Existentially generated subfields of large fields

We study subfields of large fields which are generated by infinite existentially definable subsets. We say that such subfields are existentially generated. Let $L$ be a large field of characteristic exponent $p$, and let $E\subseteq L$ be an infinite existentially generated subfield. We show that $E$ contains $L^{(p^{n})}$, the $p^{n}$-th powers in $L$, for some $n<ω$. This generalises a result of Fehm, which shows $E=L$ under the assumption that $L$ is perfect. Our method is to first study existentially generated subfields of henselian fields. Since $L$ is existentially closed in the henselian field $L((t))$, our result follows.

math.LO↗