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Universal-existential theories of fields

Part of: Connections with logic Topological fields Model theory Curves

Published online by Cambridge University Press:  03 September 2025

Sylvy Anscombe Open the ORCID record for Sylvy Anscombe [Opens in a new window]  and
Arno Fehm Open the ORCID record for Arno Fehm [Opens in a new window]
Sylvy Anscombe*
Affiliation:
Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, Paris, France
Arno Fehm
Affiliation:
Institut für Algebra, Technische Universität Dresden, Dresden, Germany
*
Corresponding author: Sylvy Anscombe, email: sylvy.anscombe@imj-prg.fr
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Abstract

We study 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.

Dedicated to the memory of Alexander Prestel (1941–2024)

Keywords

Model theoryfieldsvalued fieldsfunction fieldshenselianity

MSC classification

Primary: 03C60: Model-theoretic algebra
Secondary: 12L05: Decidability 12L12: Model theory 12J20: General valuation theory 14H05: Algebraic functions; function fields

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 69 , Issue 1 , February 2026 , pp. 95 - 124
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Edinburgh Mathematical Society.

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References

Anscombe, S., One-dimensional F-definable sets in $F(\!(t)\!)$, To appear in J. Algebra (2025), arXiv:1503.05803v2 [math.LO].Google Scholar
Anscombe, S., Dittmann, P. and Fehm, A., Axiomatizing the existential theory of $\mathbb{F}_{q}(\!(t)\!)$, Algebra & Number Theory 17(11) (2023), 2013–2032.CrossRefGoogle Scholar
Anscombe, S. and Fehm, A., The existential theory of equicharacteristic henselian valued fields, Algebra & Number Theory 10(3) (2016), 665–683.CrossRefGoogle Scholar
Anscombe, S., and Fehm, A., Characterizing diophantine henselian valuations rings and valuation ideals, Proc. Lon. Math. Soc. 115(3) (2017), 293–322.CrossRefGoogle Scholar
Anscombe, S. and Fehm, A., Interpretations of syntactic fragments of theories of fields, To appear in Israel J. Math. (2025), arXiv:2312.17616v2 [math.LO].Google Scholar
Ax, J., On the undecidability of power series fields, Proc. Amer. Math. Soc. 16(846) (1965).Google Scholar
Ax, J. and Kochen, S., Diophantine problems over local fields: III. Decidable fields, Ann. of Math. (2) 83(3) (1966), 437–456.CrossRefGoogle Scholar
Becher, K. J., Daans, N. and Dittmann, P., Uniform existential definitions of valuations in function fields in one variable. arXiv:2311.06044 [math.NT].Google Scholar
Bourbaki, N., Algebra II. Chapters 4-7 (Springer, 1990).Google Scholar
Chang, C. C. and Keisler, H. J., Model theory. Studies in Logic and the Foundations of Mathematics, 73, 3rd edition (North-Holland Publishing Co., Amsterdam, 1990).Google Scholar
Degroote, C. and Demeyer, J., Hilbert’s Tenth Problem for rational function fields over p-adic fields, J. Algebra 361 (2012), 172–187.CrossRefGoogle Scholar
Denef, J., The diophantine problem for polynomial rings and fields of rational functions, Trans. Amer. Math. Soc. 242 (1978), 391–399.CrossRefGoogle Scholar
Denef, J. and Schoutens, H.. On the decidability of the existential theory of $F_p[\![t]\!]$. In Valuation Theory and Its Applications, edited by. Kuhlmann, F.-V., Fields Inst. Commun. 33, Vol. II, pp. 43–60 (Amer. Math. Soc., Providence, RI, 2003).Google Scholar
Daans, N., Dittmann, P. and Fehm, A., Existential rank and essential dimension of diophantine sets. arXiv:2102.06941 [math.NT], 2021.Google Scholar
Dittmann, P. and Fehm, A., On the existential theory of the completions of a global field. arXiv:2401.11930 [math.LO].Google Scholar
Eisenträger, K. and Shlapentokh, A., Hilbert’s Tenth Problem over function fields of positive characteristic not containing the algebraic closure of a finite field, J. Eur. Math. Soc. 9(2103) (2017), 2138.Google Scholar
Engler, A. and Prestel, A., Valued fields. Springer Monographs in Mathematics (Springer, 2005).Google Scholar
Ershov, Y. L., Multi-Valued Fields (Springer, 2001).CrossRefGoogle Scholar
Fehm, A. and Jahnke, F., Recent progress on definability of Henselian valuations. In Ordered Algebraic Structures and Related Topics, Contemp. Math., Vol. 697, pp. 135–143 (Amer. Math. Soc., Providence, RI, 2017).CrossRefGoogle Scholar
Fried, M. D. and Jarden, M., Field Arithmetic, 3rd edition (Springer, 2008).Google Scholar
Guingona, V., Hill, C. D. and Scow, L., Characterizing model-theoretic dividing lines via collapse of generalized indiscernibles, Annals of Pure and Applied Logic 168(5) (2017), 1091–1111.CrossRefGoogle Scholar
Hess, F., An algorithm for computing isomorphisms of algebraic function fields. In Algorithmic Number Theory, Lecture Notes in Comput. Sci., Vol. 3076, pp. 263–271 (Springer-Verlag, Berlin, 2004).CrossRefGoogle Scholar
Hodges, W., A Shorter Model Theory (Cambridge University Press, 1997).Google Scholar
Koenigsmann, J., Defining transcendentals in function fields, J. Symb. Logic 67(3) (2002), 947–956.CrossRefGoogle Scholar
Koenigsmann, J., Undecidability in number theory. In Model Theory in Algebra, Analysis and Arithmetic, Lecture Notes in Math., Fond. CIME/CIME Found. Subser., Vol. 2111, pp. 159–195 (Springer, Heidelberg, 2014).Google Scholar
Kartas, K., Diophantine problems over tamely ramified fields, J. Algebra 617 (2023), 127–159.CrossRefGoogle Scholar
Kartas, K., Valued fields with a total residue map, J. Math. Logic 24(3) (2024), 14.CrossRefGoogle Scholar
Kuhlmann, F.-V., Elementary properties of power series fields over finite fields, J. Symb. Logic 66(2) (2001), 771–791.CrossRefGoogle Scholar
Miller, R., Poonen, B., Schoutens, H. and Shlapentokh, A., A computable functor from graphs to fields, J. Symb. Logic 83(1) (2018), 326–348.CrossRefGoogle Scholar
Moret-Bailly, L., Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real and p-adic fields, J. Reine Angew. Math. 587 (2005), 77–143.CrossRefGoogle Scholar
Onay, G., $\mathbb{F}_p(\!(X)\!)$ is Decidable as a Module Over the Ring of Additive Polynomials. arXiv:1806.03123 [math.LO].Google Scholar
Pheidas, T., Hilbert’s Tenth Problem for fields of rational functions over finite fields, Invent. Math. 103(1) (1991), 1–8.CrossRefGoogle Scholar
Post, E. L., Recursively enumerable sets of positive integers and their decision problems, Bull. Amer. Math. Soc. 50 (1944), 284–316.CrossRefGoogle Scholar
Prestel, A. and Delzell, C. N., Mathematical Logic and Model Theory (Springer, 2011).CrossRefGoogle Scholar
Rosenlicht, M., Automorphisms of function fields, Trans. Amer. Math. Soc. 79(1) (1955), 1–11.CrossRefGoogle Scholar
Schilling, O. F. G., Automorphisms of fields of formal power series, Bull. Amer. Math. Soc. 50(12) (1944), 892–901.CrossRefGoogle Scholar
Soare, R.I., Turing Computability. Theory and Applications (Springer-Verlag, Berlin, 2016), Theory Appl. Comput.Google Scholar
Stichtenoth, H., Algebraic Function Fields and Codes, Grad. Texts in Math., Vol. 254, pp. xiv+355 (Springer-Verlag, Berlin, 2009).Google Scholar
Tent, K. and Ziegler, M., A course in model theory. Lect. Notes Log. Association for Symbolic Logic, La Jolla, Vol. 40 (Cambridge University Press, Cambridge, 2012).Google Scholar
Tyrrell, B., Undecidability in some field theories. Dissertation, Oxford, 2023.Google Scholar
Videla, C., Hilbert’s Tenth Problem for rational function fields in characteristic 2, Proc. Amer. Math. Soc. 120(1) (1994), 249–253.Google Scholar
Vollprecht, A., Model theory of rings and fields finitely generated over local fields. Dissertation, Dresden, 2025.Google Scholar