19 citations · 59 across the 19 of their papers we have counts for
19 papers
On the existential theory of the completions of a global field
Philip Dittmann, Arno Fehm
We discuss the common existential theory of all or almost all completions of a global function field.
Uniform existential definitions of valuations in function fields in one variable
Karim Johannes Becher, Nicolas Daans, Philip Dittmann
We study function fields of curves over a base field which is either a global field or a large field having a separable field extension of degree divisible by . We show that…
Ax-Kochen-Ershov principles for finitely ramified henselian fields
Sylvy Anscombe, Philip Dittmann, Franziska Jahnke
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 t…
Two examples concerning existential undecidability in fields
Philip Dittmann
We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answerin…
When is the étale open topology a field topology?
Philip Dittmann, Erik Walsberg, Jinhe Ye
We investigate the following question: Given a field , when is the étale open topology induced by a field topology? On the positive side, when is the fractio…
Definable valuations on ordered fields
Philip Dittmann, Franziska Jahnke, Lothar Sebastian Krapp +1
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, o…
Axiomatizing the existential theory of Fq((t))
Sylvy Anscombe, Philip Dittmann, Arno Fehm
We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singular…
Existential rank and essential dimension of diophantine sets
Nicolas Daans, Philip Dittmann, Arno Fehm
We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points as…
Odoni's conjecture on arboreal Galois representations is false
Philip Dittmann, Borys Kadets
Suppose is a polynomial. The absolute Galois group of acts on the preimage tree of under . The resulting homomorphism $ϕ_f: \mathrm{Gal}_K \to…
Characterizing finitely generated fields by a single field axiom
Philip Dittmann, Florian Pop
We resolve the strong Elementary Equivalence versus Isomorphism Problem for finitely generated fields. That is, we show that for every field in this class there is a first-order se…
Non-definability of rings of integers in most algebraic fields
Philip Dittmann, Arno Fehm
We show that the set of algebraic extensions of in which or the ring of integers are definable is meager in the set of all algebraic e…
A class of fields with a restricted model completeness property
Philip Dittmann, Dion Leijnse
We introduce and study a natural class of fields in which certain first-order definable sets are existentially definable, and characterise this class by a number of equivalent cond…
Denseness results in the theory of algebraic fields
Sylvy Anscombe, Philip Dittmann, Arno Fehm
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…
The dimension growth conjecture, polynomial in the degree and without logarithmic factors
Wouter Castryck, Raf Cluckers, Philip Dittmann +1
We address Heath-Brown's and Serre's dimension growth conjecture (proved by Salberger), when the degree grows. Recall that Salberger's dimension growth results give bounds of t…
A p-adic analogue of Siegel's Theorem on sums of squares
Sylvy Anscombe, Philip Dittmann, Arno Fehm
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.…
Approximation theorems for spaces of localities
Sylvy Anscombe, Philip Dittmann, Arno Fehm
The classical Artin--Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequival…
Defining Subrings in Finitely Generated Fields of All Characteristics
Philip Dittmann
We give a construction of a large first-order definable family of subrings of finitely generated fields of any characteristic. We deduce that for any such there exists a fi…
Irreducibility of Polynomials over Global Fields is Diophantine
Philip Dittmann
Given a global field and a positive integer , we present a diophantine criterion for a polynomial in one variable of degree over not to have any root in . This st…
Ultraproducts as a tool for first-order inexpressibility in the finite and infinite
Philip Dittmann
Ultraproducts are a well-known tool in the classical model theory of first-order logic. We explore their uses in the context of finite model theory.