Model theory of the Frobenius on the Witt vectors

We give axiomatizations and prove quantifier elimination theorems for first-order theories of unramified valued fields with an automorphism having a close interaction with the valuation. We achieve an analogue of the classical Ostrowski theory of pseudoconvergence. In the outstanding case of Witt vectors with their Frobenius map, we use the ∂-ring formalism from Joyal.

[1]  Thomas Scanlon,et al.  Quantifier Elimination for the Relative Frobenius , 2000 .

[2]  Z. Chatzidakis.  A Survey on the Model Theory of Difference Fields , 2000 .

[3]  Jan Denef,et al.  p-adic semi-algebraic sets and cell decomposition. , 1986 .

[4]  Michiel Hazewinkel,et al.  Formal Groups and Applications , 1978 .

[5]  Luc Bélair,et al.  L'automorphisme de Frobenius des vecteurs de Witt , 2000 .

[6]  D. Marker.  Model theory : an introduction , 2002 .

[7]  Thomas Scanlon,et al.  A model complete theory of valued D-fields , 2000, Journal of Symbolic Logic.

[8]  Luc Bélair,et al.  Types dans les corps valués munis d'applications coefficients , 1999 .

[9]  Serban A. Basarab,et al.  Relative Elimination of Quantifiers for Henselian Valued Fields , 1991, Ann. Pure Appl. Log..

[10]  Paulo Ribenboim,et al.  Théorie des valuations , 1964 .

[11]  Jean-Louis Krivine,et al.  Éléments de logique mathématique : théorie des modèles , 1967 .

[12]  T. Browning,et al.  Local Fields , 2008 .

[13]  Peter Roquette,et al.  Formally P-Adic Fields , 1984 .

[14]  G. Whaples.  Galois cohomology of additive polynomial and $n$-${\rm th}$power mappings of fields , 1957 .

[15]  Chapter 9 Hensel ’ s Lemma 9 . 1 Equivalent forms of Hensel ’ s Lemma , 2022 .

[16]  J. Neukirch,et al.  The Theory of Valuations , 1999 .

[18]  Moshe Jarden,et al.  THE ELEMENTARY THEORY OF FINITE FIELDS , 2004 .

[19]  A. Duval.  Lemmes de Hensel et factorisation formelle pour les opérateurs aux différences , 1983 .

[20]  Hirotaka Kikyo,et al.  Model companions of theories with an automorphism , 1998, Journal of Symbolic Logic.

[21]  Simon Kochen,et al.  The model theory of local fields , 1975 .

[22]  Angus Macintyre,et al.  Generic Automorphisms of Fields , 1997, Ann. Pure Appl. Log..

[23]  M. Marshall,et al.  Valuation Theory and Its Applications, Volume II , 2003 .

[24]  Irving Kaplansky,et al.  Maximal fields with valuations, II , 1942 .

[25]  Franz-Viktor Kuhlmann.  Quantifier elimination for henselian fields relative to additive and multiplicative congruences , 1994 .

[26]  S. Kochen,et al.  Diophantine Problems Over Local Fields I , 1965 .

[27]  A. Buium.  Geometry of $p$-jets , 1996 .