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Generic derivations on algebraically bounded structures (2310.20511v4)

Published 31 Oct 2023 in math.LO and math.AC

Abstract: Let K be an algebraically bounded structure and T be its theory. If T is model complete, then the theory of K endowed with a derivation, denoted by $T{\delta}$, has a model completion. Additionally, we prove that if the theory T is stable/NIP then the model completion of $T{\delta}$ is also stable/NIP. Similar results hold for the theory with several derivations, either commuting or non-commuting.

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References (55)
  1. Matthias Aschenbrenner, Lou Dries and Joris Hoeven “Asymptotic Differential Algebra and Model Theory of Transseries” Princeton University Press DOI: 10.1515/9781400885411
  2. Lenore Blum “Differentially closed fields: a model-theoretic tour” In Contributions to Algebra Academic Press, 1977, pp. 37–61 DOI: 10.1016/B978-0-12-080550-1.50009-3
  3. Angela Borrata “Model Theory of Tame Pairs of Closed Ordered Differential Fields”, 2021
  4. Thomas Brihaye, Christian Michaux and Cédric Rivière “Cell decomposition and dimension function in the theory of closed ordered differential fields” In Ann. Pure Appl. Logic 159.1-2, 2009, pp. 111–128 DOI: 10.1016/j.apal.2008.09.029
  5. Quentin Brouette, Pablo Cubides Kovacsics and Françoise Point “Strong density of definable types and closed ordered differential fields” In J. Symb. Log. 84.3, 2019, pp. 1099–1117 DOI: 10.1017/jsl.2018.88
  6. Zoé Chatzidakis “Model Theory of Fields with Operators – a Survey” In Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics De Gruyter, 2015, pp. 91–114
  7. “Generic structures and simple theories” In Annals of Pure and Applied Logic 95.1, 1998, pp. 71–92 DOI: 10.1016/S0168-0072(98)00021-9
  8. Pablo Cubides Kovacsics and Françoise Point “Topological fields with a generic derivation” In Ann. Pure Appl. Logic 174.3, 2023, pp. Paper No. 103211\bibrangessep38 DOI: 10.1016/j.apal.2022.103211
  9. Lou Dries “Model theory of fields: Decidability, and bounds for polynomial ideals” Advised by D. van Dalen., 1978
  10. Lou Dries “Dimension of definable sets, algerbraic boundedness and Henselian fields” In Annals of Pure and Applied Logic 45, 1989, pp. 189–209
  11. “Bounds in the theory of polynomial rings over fields. A nonstandard approach” In Invent. Math. 76.1, 1984, pp. 77–91 DOI: 10.1007/BF01388493
  12. Arno Fehm “Subfields of ample fields. Rational maps and definability” In Journal of Algebra 323.6, 2010, pp. 1738–1744 DOI: 10.1016/j.jalgebra.2009.11.037
  13. Antongiulio Fornasiero “Dimensions, matroids, and dense pairs of first-order structures” In Annals of Pure and Applied Logic 162.7, 2011, pp. 514–543 DOI: 10.1016/j.apal.2011.01.003
  14. “Generic derivations on o-minimal structures” In Journal of Mathematical Logic, 2020 DOI: 10.1142/S0219061321500070
  15. James Freitag, Omar León Sánchez and Wei Li “Effective definability of Kolchin polynomials” In Proc. Amer. Math. Soc. 148.4, 2020, pp. 1455–1466 DOI: 10.1090/proc/14869
  16. “Topological differential fields” In Ann. Pure Appl. Logic 161.4, 2010, pp. 570–598 DOI: 10.1016/j.apal.2009.08.001
  17. “Topological differential fields and dimension functions” In J. Symbolic Logic 77.4, 2012, pp. 1147–1164 DOI: 10.2178/jsl.7704050
  18. “Geometrical axiomatization for model complete theories of differential topological fields” In Notre Dame J. Formal Logic 47.3, 2006, pp. 331–341 DOI: 10.1305/ndjfl/1163775440
  19. “Lectures on algebraic model theory” Fields Institute Monographs, 2002 DOI: 10.1007/978-0-8218-2706-2
  20. Will Johnson “Forking and dividing in fields with several orderings and valuations” In J. Math. Log. 22.1, 2022, pp. Paper No. 2150025\bibrangessep43 DOI: 10.1142/S0219061321500252
  21. “Curve-excluding fields”, 2023 arXiv:2303.06063v1
  22. “A note on geometric theories of fields”, 2022 arXiv:2208.00586
  23. “Schlanke Körper (Slim fields)” In The Journal of Symbolic Logic 75.2 Cambridge University Press, 2010, pp. 481–500 DOI: 10.2178/jsl/1268917491
  24. E.R. Kolchin “Differential algebra and algebraic groups”, Pure and Applied Mathematics, Vol. 54 Academic Press, New York-London, 1973, pp. xviii+446
  25. Serge Lang “Algebra” 211, Graduate Texts in Mathematics Springer-Verlag, New York, 2002, pp. xvi+914 DOI: 10.1007/978-1-4613-0041-0
  26. Omar León Sánchez “Algebro-geometric axioms for DCF0,msubscriptDCF0𝑚{\rm DCF}_{0,m}roman_DCF start_POSTSUBSCRIPT 0 , italic_m end_POSTSUBSCRIPT” In Fund. Math. 243.1, 2018, pp. 1–8 DOI: 10.4064/fm228-11-2017
  27. Omar León Sánchez and Marcus Tressl “Differentially Large Fields”, 2020 arXiv:2005.00888
  28. Omar León Sánchez and Marcus Tressl “On ordinary differentially large fields”, 2023 arXiv:2307.12977
  29. David Marker, Margit Messmer and Anand Pillay “Model Theory of Fields”, Lecture Notes in Logic Cambridge University Press, 2017 DOI: 10.1017/9781316716991
  30. Tracey McGrail “The model theory of differential fields with finitely many commuting derivations” In The Journal of Symbolic Logic 65.2 Cambridge University Press, 2000, pp. 885–913
  31. Shezad Mohamed “The uniform companion for large fields with free operators in characteristic zero”, 2023 arXiv:2311.01856
  32. “Pseudo T-closed fields”, 2023 arXiv:2304.10433 [math.LO]
  33. Rahim Moosa “Six lectures on model theory and differential-algebraic geometry”, 2022 arXiv:2210.16684
  34. “Model theory of fields with free operators in characteristic zero” In Journal of Mathematical Logic 42.02, 2014 DOI: 10.1142/S0219061314500093
  35. David Pierce “Differential forms in the model theory of differential fields” In J. Symbolic Logic 68.3, 2003, pp. 923–945 DOI: 10.2178/jsl/1058448448
  36. David Pierce “Fields with several commuting derivations” In The Journal of Symbolic Logic 79.1 [Association for Symbolic Logic, Cambridge University Press], 2014, pp. 1–19 JSTOR: 43303717
  37. “A note on the axioms for differentially closed fields of characteristic zero” In J. Algebra 204.1, 1998, pp. 108–115 DOI: 10.1006/jabr.1997.7359
  38. Anand Pillay “An introduction to stability theory” Mineola, NY: Dover Publications, 2008
  39. Françoise Point “Ensembles définissables dans les corps ordonnés différentiellement clos” In Comptes Rendus. Mathématique 349.17–18, 2011, pp. 929–933 DOI: 10.1016/j.crma.2011.08.003
  40. Cédric Rivière “The model theory of m𝑚mitalic_m-ordered differential fields” In MLQ Math. Log. Q. 52.4, 2006, pp. 331–339 DOI: 10.1002/malq.200510037
  41. Cédric Rivière “The theory of closed ordered differential fields with m𝑚mitalic_m commuting derivations” In C. R. Math. Acad. Sci. Paris 343.3, 2006, pp. 151–154 DOI: 10.1016/j.crma.2006.06.019
  42. Cédric Rivière “Further notes on cell decomposition in closed ordered differential fields” In Ann. Pure Appl. Logic 159.1-2, 2009, pp. 100–110 DOI: 10.1016/j.apal.2008.11.002
  43. Abraham Robinson “On the concept of a differentially closed field” In Bull. Res. Council Israel Sect. F 8F, 1959, pp. 113–128
  44. Gerald E Sacks “Saturated Model Theory” World Scientific, 2009 DOI: 10.1142/6974
  45. Thomas Scanlon “Model Theory of Valued D-Fields”, 1997
  46. Thomas Scanlon “A Model Complete Theory of Valued D-Fields” In The Journal of Symbolic Logic 65.4 [Association for Symbolic Logic, Cambridge University Press], 2000, pp. 1758–1784 DOI: 10.2307/2695074
  47. Hans Schoutens “The use of ultraproducts in commutative algebra” 1999, Lecture Notes in Mathematics Springer-Verlag, Berlin, 2010, pp. x+204 DOI: 10.1007/978-3-642-13368-8
  48. Saharon Shelah “Classification Theory: and the Number of Non-Isomorphic Models” Elsevier, 1990
  49. Pierre Simon “A guide to NIP theories”, 2015 URL: https://www.normalesup.org/~simon/NIP_guide.pdf
  50. Michael F. Singer “The model theory of ordered differential fields” In J. Symbolic Logic 43.1, 1978, pp. 82–91 DOI: 10.2307/2271951
  51. Michael F. Singer “Model Theory of Partial Differential Fields: From Commuting to Noncommuting Derivations” In Proceedings of the American Mathematical Society 135.6 American Mathematical Society, 2007, pp. 1929–1934 JSTOR: 20534779
  52. Marcus Tressl “The uniform companion for large differential fields of characteristic 0” In Trans. Amer. Math. Soc. 357.10, 2005, pp. 3933–3951 DOI: 10.1090/S0002-9947-05-03981-4
  53. Carol Wood “The Model Theory of Differential Fields of Characteristic p≠0𝑝0p\neq 0italic_p ≠ 0” In Proceedings of the American Mathematical Society 40.2 American Mathematical Society, 1973, pp. 577–584 JSTOR: 2039417
  54. Yoav Yaffe “Model completion of Lie differential fields” In Annals of Pure and Applied Logic 107.1, 2001, pp. 49–86 DOI: 10.1016/S0168-0072(00)00025-7
  55. “Commutative Algebra” 1, Graduate Texts in Mathematics Berlin, Heidelberg: Springer Berlin Heidelberg, 1960 DOI: 10.1007/978-3-662-29244-0
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