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THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

  • OMAR LEÓN SÁNCHEZ (a1) and RAHIM MOOSA (a2)

Abstract

A model companion is shown to exist for the theory of partial differential fields of characteristic zero equipped with free operators that commute with the derivations. The free operators here are those introduced in [R. Moosa and T. Scanlon, Model theory of fields with free operators in characteristic zero, Journal of Mathematical Logic 14(2), 2014]. The proof relies on a new lifting lemma in differential algebra: a differential version of Hensel’s Lemma for local finite algebras over differentially closed fields.

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[1] Bustamante, T., Théorie de modèles des corps différentiellement clos avec un automorphisme génerique , PhD Thesis, Université Paris 7, 2005.
[2] Chatzidakis, Z. and Hrushovski, E., Model theory of difference fields . Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 29973071.
[3] Kolchin, E. R., Differential Algebra and Algebraic Groups, Pure and Applied Mathematics, vol. 54, Academic Press, New York-London, 1973.
[4] Kolchin, E. R., Differential Algebraic Groups, Pure and Applied Mathematics, vol. 114, Academic Press, 1985.
[5] León Sánchez, O., Geometric axioms for differentially closed fields with several commuting derivations . Journal of Algebra, vol. 362 (2012), pp. 107116. Corrigendum, pp. 332–334.
[6] León Sánchez, O., On the model companion of partial differential fields with an automorphism. Israel Journal of Mathematics , to appear.
[7] Moosa, R. and Scanlon, T., Jet and prolongation spaces . Journal de l’Institut de Mathématiques de Jussieu, vol. 9 (2010), no. 2, pp. 391430.
[8] Moosa, R. and Scanlon, T., Model theory of fields with free operators in characteristic zero . Journal of Mathematical Logic, vol. 14 (2014), no. 2, pp. 1450009, 43.
[9] Rosenfeld, Azriel, Specializations in differential algebra . Transactions of the American Mathematical Society, vol. 90 (1959), pp. 394407.
[10] Scanlon, T., A model complete theory of valued D-fields , this Journal, vol. 65 (2000), no. 4, pp. 17581784.

Keywords

THE MODEL COMPANION OF DIFFERENTIAL FIELDS WITH FREE OPERATORS

  • OMAR LEÓN SÁNCHEZ (a1) and RAHIM MOOSA (a2)

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