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Philip Dittmann

Publications and source records attributed to Philip Dittmann.

At least 19 recordsLinked to original sources
arXiv2026

Existential theories of henselian valued fields under a formal smoothness assumption

We study existential theories of henselian valued fields of positive characteristic with parameters from a trivially valued subfield. Compared to previous work, we relax perfectness and separability assumptions, and instead work with the weaker algebraic hypothesis of formal smoothness over the parameter field, which we discuss in detail in our setting. Assuming a weak consequence of resolution of singularities, which was already used in previous work, we obtain an axiomatisation of the existential theory of such a valued field in terms of the existential theory of the residue field, both over the same parameter field. This result has natural applications to asymptotic theories of completions of function fields of curves. We work these out in detail for the case of function fields over fairly general pseudo-algebraically closed fields, where we obtain decidability of the sets of universal/existential sentences holding in all completions or all but finitely many completions, respectively.

math.LO↗
arXiv2026

Composition Ax-Kochen/Ershov principles and tame fields of mixed characteristic

We study in which settings we have a composition AKE principle, i.e. when the theory of the coarsening $(K,w)$ and the theory of the induced valuation $(Kw,\overline{v})$ determine the theory of the composition $(K,v)$. We show that this is the case when $(K,w)$ is tame of equal characteristic, and provide counterexamples in mixed characteristic. We further show that, for a tame field of mixed characteristic, the theory of the valued field cannot, in general, be determined solely by the theories of its underlying field, its residue field, and its value group.

math.LO↗
arXiv2024

Universally defining subrings in function fields

We establish that all rings of $S$-integers are universally definable in function fields in one variable over certain ground fields including global and non-archimedean local fields. That is, we show that the complement of such a ring of $S$-integers is always a diophantine set. As a technical tool, we use a reciprocity exact sequence for quadratic Witt groups in function fields over almost arbitrary base fields (of any characteristic), which is new and of potentially independent interest.

math.NT↗
arXiv2023

Uniform existential definitions of valuations in function fields in one variable

We study function fields of curves over a base field $K$ which is either a global field or a large field having a separable field extension of degree divisible by $4$. We show that, for any such function field, Hilbert's 10th Problem has a negative answer, the valuation rings containing $K$ are uniformly existentially definable, and finitely generated integrally closed $K$-subalgebras are definable by a universal-existential formula. In order to obtain these results, we develop further the usage of local-global principles for quadratic forms in function fields to definability of certain subrings. We include a first systematic presentation of this general method, without restriction on the characteristic.

math.NT↗
arXiv2023

Characterising local fields of positive characteristic by Galois theory and the Brauer group

We show that each local field $\mathbb{F}_q((t))$ of characteristic $p > 0$ is characterised up to isomorphism within the class of all fields of imperfect exponent at most $1$ by (certain small quotients of) its absolute Galois group together with natural axioms concerning the $p$-torsion of its Brauer group. This complements previous work by Efrat-Fesenko, who analysed fields whose absolute Galois group is isomorphic to that of a local field of characteristic $p$.

math.NT↗
arXiv2023

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↗
arXiv2022

Two examples concerning existential undecidability in fields

We construct an existentially undecidable complete discretely valued field of mixed characteristic with existentially decidable residue field and decidable algebraic part, answering a question by Anscombe-Fehm in a strong way. Along the way, we construct an existentially decidable field of positive characteristic with an existentially undecidable finite extension, modifying a construction due to Kesavan Thanagopal.

math.LO↗
arXiv2022

When is the \'etale open topology a field topology?

We investigate the following question: Given a field $K$, when is the \'etale open topology $\mathcal{E}_K$ induced by a field topology? On the positive side, when $K$ is the fraction field of a local domain $R\neq K$, using a weak form of resolution of singularities due to Gabber, we show that $\mathcal{E}_K$ agrees with the $R$-adic topology when $R$ is quasi-excellent and henselian. Various pathologies appear when dropping the quasi-excellence assumption. For locally bounded field topologies, we introduce the notion of generalized t-henselianity (gt-henselianity) following Prestel and Ziegler. We establish the following: For a locally bounded field topology $\tau$, the \'etale open topology is induced by $\tau$ if and only if $\tau$ is gt-henselian and some non-empty \'etale image is $\tau$-bounded open. On the negative side, we obtain that for a pseudo-algebraically closed field $K$, $\mathcal{E}_K$ is never induced by a field topology.

math.LO↗
arXiv2022

Definable valuations on ordered fields

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 and compare definability in both languages and show the following contrary results: while there are convex valuations that are definable in the language $\mathcal{L}_{\mathrm{or}}$ but not in the language $\mathcal{L}_{\mathrm{r}}$, any $\mathcal{L}_{\mathrm{or}}$-definable henselian valuation is already $\mathcal{L}_{\mathrm{r}}$-definable. To prove the latter, we show that the value group and the ordered residue field of an ordered henselian valued field are stably embedded (as an ordered abelian group, respectively as an ordered field). Moreover, we show that in almost real closed fields any $\mathcal{L}_{\mathrm{or}}$-definable valuation is henselian.

math.LO↗
arXiv2022

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↗
arXiv2020

Odoni's conjecture on arboreal Galois representations is false

Suppose $f \in K[x]$ is a polynomial. The absolute Galois group of $K$ acts on the preimage tree $\mathrm{T}$ of $0$ under $f$. The resulting homomorphism $\phi_f: \mathrm{Gal}_K \to \mathrm{Aut} \mathrm{T}$ is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields $K$ there exists a polynomial $f$ for which $\phi_f$ is surjective. We show that this conjecture is false.

math.NT↗
arXiv2020

Characterizing finitely generated fields by a single field axiom

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 sentence which characterizes this field within the class up to isomorphism. Our solution is conditional on resolution of singularities in characteristic two and unconditional in all other characteristics.

math.LO↗
arXiv2019

A class of fields with a restricted model completeness property

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 conditions. We show that global fields belong to this class, and in particular obtain a number of new existential (or diophantine) predicates over global fields.

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↗