Philip Dittmann

Research

I study fields (in the sense of algebra) from a variety of perspectives, mostly motivated by model theory and Hilbert's Tenth Problem. In particular, I have worked on henselian valued fields and definability questions in finitely generated fields.

Publications & Preprints

• Existential theories of henselian valued fields under a formal smoothness assumption

  P. Dittmann. Preprint, 2026.  hal-05526181

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

  M. Ketelsen, with an appendix by P. Dittmann. Preprint, 2026.  arXiv:2601.05790

• Universally defining subrings in function fields

  N. Daans, P. Dittmann. Accepted for publication in Journal für die reine und angewandte Mathematik.  arXiv:2404.02749

• On the existential theory of the completions of a global field

  P. Dittmann, A. Fehm. Preprint, 2024.  arXiv:2401.11930

• Uniform existential definitions of valuations in function fields in one variable

  K. J. Becher, N. Daans, P. Dittmann. Preprint, 2023.  arXiv:2311.06044

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

  P. Dittmann. Compositio Mathematica 161(8):2136–2153, 2025.  arXiv:2309.05398

• Ax–Kochen–Ershov principles for finitely ramified henselian fields

  S. Anscombe, P. Dittmann, F. Jahnke. Transactions of the American Mathematical Society 377(12):8963–8988, 2024.  arXiv:2305.12145

• Two examples concerning existential undecidability in fields

  P. Dittmann. Journal of Symbolic Logic 90(2):552–563, 2025.  arXiv:2211.01775

• When is the étale open topology a field topology?

  P. Dittmann, E. Walsberg, J. Ye. Israel Journal of Mathematics 269:67–102, 2025.  arXiv:2208.02398

• Definable valuations on ordered fields

  P. Dittmann, F. Jahnke, L. S. Krapp, S. Kuhlmann. Model Theory 2(1):101–120, 2023.  arXiv:2206.15301

• Axiomatizing the existential theory of 𝔽ₚ((t))

  S. Anscombe, P. Dittmann, A. Fehm. Algebra & Number Theory 17(11):2013–2032, 2023.  arXiv:2205.05438

• Existential rank and essential dimension of diophantine sets

  N. Daans, P. Dittmann, A. Fehm. Preprint, 2021.  arXiv:2102.06941

• Odoni’s conjecture on arboreal Galois representations is false

  P. Dittmann, B. Kadets. Proceedings of the American Mathematical Society 150(8):3335–3343, 2022.  arXiv:2012.03076

• Characterizing finitely generated fields by a single field axiom

  P. Dittmann, F. Pop. Annals of Mathematics 198(3):1203–1227, 2023.  arXiv:2012.01307

• Non-definability of rings of integers in most algebraic fields

  P. Dittmann, A. Fehm. Notre Dame Journal of Formal Logic 62(3):589–592, 2021.  arXiv:2011.14367

• Galois groups of large simple fields

  A. Pillay, E. Walsberg, with an appendix by P. Dittmann. Model Theory 2(2):357–380, 2023.  arXiv:2011.10018

• A class of fields with a restricted model completeness property

  P. Dittmann, D. Leijnse. Journal of Symbolic Logic 86(2):701–708, 2021.  arXiv:1911.03202

• Denseness results in the theory of algebraic fields

  S. Anscombe, P. Dittmann, A. Fehm. Annals of Pure and Applied Logic 172(8), 2021.  arXiv:1909.12188

• The dimension growth conjecture, polynomial in the degree and without logarithmic factors

  W. Castryck, R. Cluckers, P. Dittmann, K. H. Nguyen. Algebra & Number Theory 14(4):2261–2294, 2020.  arXiv:1904.13109

• A p-adic analogue of Siegel’s theorem on sums of squares

  S. Anscombe, P. Dittmann, A. Fehm. Mathematische Nachrichten 293(8):1434–1451, 2020.  arXiv:1904.03466

• Approximation theorems for spaces of localities

  S. Anscombe, P. Dittmann, A. Fehm. Mathematische Zeitschrift 296(3):1471–1499, 2020.  arXiv:1901.02632

• Irreducibility of polynomials over global fields is diophantine

  P. Dittmann. Compositio Mathematica 154(4):761–772, 2018.  arXiv:1601.07829

CV

I was previously a postdoctoral researcher at TU Dresden, at the MSRI/SLMath in Berkeley, and at KU Leuven. I was a student at Oxford—where I wrote my thesis A model-theoretic approach to the arithmetic of global fields—, Cambridge, and TU Darmstadt.