Advertisement

Selecta Mathematica

, 25:57 | Cite as

A primitive element theorem for fields with commuting derivations and automorphisms

  • Gleb PogudinEmail author
Article
  • 45 Downloads

Abstract

We establish a Primitive Element Theorem for fields equipped with several commuting operators such that each of the operators is either a derivation or an automorphism. More precisely, we show that for every extension \(F \subset E\) of such fields of zero characteristic such that
  • E is generated over F by finitely many elements using the field operations and the operators,

  • every element of E satisfies a nontrivial equation with coefficient in F involving the field operations and the operators,

  • the action of the operators on E is irredundant

there exists an element \(a \in E\) such that E is generated over F by a using the field operations and the operators. This result generalizes the Primitive Element Theorems by Kolchin and Cohn in two directions simultaneously: we allow any numbers of derivations and automorphisms and do not impose any restrictions on the base field F.

Keywords

Primitive element Differential field Difference field Fields with operators 

Mathematics Subject Classification

Primary 12H05 12H05 Secondary 12F99 

Notes

Acknowledgements

The author is grateful to Lei Fu, Alexey Ovchinnikov, Thomas Scanlon, and the referee for their suggestions and helpful discussions. This work has been partially supported by NSF Grants CCF-1564132, CCF-1563942, DMS-1853482, DMS-1853650, and DMS-1760448, by PSC-CUNY Grants #69827-0047 and #60098-0048.

References

  1. 1.
    Andrews, G .E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1999).  https://doi.org/10.1017/CBO9781107325937 CrossRefGoogle Scholar
  2. 2.
    Bélair, L.: Approximation for Frobenius algebraic equations in Witt vectors. J. Algebra 321(9), 2353–2364 (2009).  https://doi.org/10.1016/j.jalgebra.2009.01.021 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bell, D.J., Lu, X.Y.: Differential algebraic control theory. IMA J. Math. Control Inf. 9(4), 361–383 (1992).  https://doi.org/10.1093/imamci/9.4.361 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blossier, T., Hardouin, C., Martin-Pizarro, A.: Sur les automorphismes bornés de corps munis d’opérateurs. Math. Res. Lett. 24(4), 955–978 (2017).  https://doi.org/10.4310/MRL.2017.v24.n4.a2 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brouette, Q., Point, F.: On differential Galois groups of strongly normal extensions. Math. Logic Q. 64(3), 155–169 (2018).  https://doi.org/10.1002/malq.201600098 MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chatzidakis, Z.: Model theory of fields with operators—a survey. In : Villaveces A, Kossak R, Kontinen J, Hirvonen Å (eds) Logic Without Borders—Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, pp. 91–114. (2015).  https://doi.org/10.1515/9781614516873.91
  7. 7.
    Chatzidakis, Z., Hrushovski, E.: Model theory of difference fields. Trans. Am. Math. Soc. 351, 2997–3071 (1999).  https://doi.org/10.1090/S0002-9947-99-02498-8 MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cluzeau, T., Hubert, E.: Resolvent representation for regular differential ideals. Appl. Algebra Eng. Commun. Comput. 13(5), 395–425 (2003).  https://doi.org/10.1007/s00200-002-0110-4 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cluzeau, T., Hubert, E.: Probabilistic algorithms for computing resolvent representations of regular differential ideals. Appl. Algebra Eng. Commun. Comput. 19(5), 365–392 (2008).  https://doi.org/10.1007/s00200-008-0079-8 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cohn, R.M.: Difference Algebra. Interscience Publishers, Geneva (1965)zbMATHGoogle Scholar
  11. 11.
    D’Alfonso, L., Jeronimo, G., Solerno, P.: Quantitative aspects of the generalized differential Lüroth’s theorem. J. Algebra 507, 547–570 (2018).  https://doi.org/10.1016/j.jalgebra.2018.01.050 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fliess, M.: Generalized controller canonical form for linear and nonlinear dynamics. IEEE Trans. Autom. Control 35(9), 994–1001 (1990).  https://doi.org/10.1109/9.58527 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Freitag, J., Li, W.: Simple Differential Field Extensions and Effective Bounds. Lecture Notes in Computer Science, vol. 9582, pp. 343–357. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-32859-1_29 CrossRefzbMATHGoogle Scholar
  14. 14.
    Gao, X., Van der Hoeven, J., Yuan, C., Zhang, G.: Characteristic set method for differential-difference polynomial systems. J. Symb. Comput. 44(9), 1137–1163 (2009).  https://doi.org/10.1016/j.jsc.2008.02.010 MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hardouin, C., Singer, M.F.: Differential Galois theory of linear difference equations. Math. Ann. 342(2), 333–377 (2008).  https://doi.org/10.1007/s00208-008-0238-z MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kamensky, M.: Tannakian formalism over fields with operators. Int. Math. Res. Notices 2013(24), 5571–5622 (2013).  https://doi.org/10.1093/imrn/rns190 MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kolchin, E.R.: Extensions of differential fileds. Ann. Math. 43(4), 724–729 (1942)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kolchin, E.R.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)zbMATHGoogle Scholar
  19. 19.
    Kondratieva, M.V., Levin, A.B., Mikhalev, A.V., Pankratiev, E.V.: Differential and Difference Dimension Polynomials. Springer, Dordrecht (2010)zbMATHGoogle Scholar
  20. 20.
    Levin, A.B. : Multivariate difference-differential dimension polynomials and new invariants of difference-differential field extensions. In: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ISSAC ’13, pp. 267–274, (2013).  https://doi.org/10.1145/2465506.2465521
  21. 21.
    Levin, A.B.: Difference Algebra. Springer, Dordrecht (2008).  https://doi.org/10.1007/978-1-4020-6947-5 CrossRefzbMATHGoogle Scholar
  22. 22.
    Loos, R.: Computing in Algebraic Extensions, pp. 173–187. Springer, Vienna (1983).  https://doi.org/10.1007/978-3-7091-7551-4_12 CrossRefGoogle Scholar
  23. 23.
    Marker, D.: Chapter 2: Model Theory of Differential Fields, of Lecture Notes in Logic, vol. 5, pp. 38–113. Springer, Berlin, (1996). https://projecteuclid.org/euclid.lnl/1235423156
  24. 24.
    Medina, R.F.B.: Differentially closed fields of characteristic zero with a generic automorphism. Rev. de Mat. Teor. y Apl. 14(1), 81–100 (2007).  https://doi.org/10.15517/rmta.v14i1.282 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Miller, R., Ovchinnikov, A., Trushin, D.: Computing constraint sets for differential fields. J. Algebra 407, 316–357 (2014).  https://doi.org/10.1016/j.jalgebra.2014.02.032 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Moosa, R., Scanlon, T.: Jet and prolongation spaces. J. Inst. Math. Jussieu 9(2), 391–430 (2010).  https://doi.org/10.1017/S1474748010000010 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Moosa, R., Scanlon, T.: Model theory of fields with free operators in characteristic zero. J. Math. Logic 14(2), 1450009 (2014).  https://doi.org/10.1142/S0219061314500093 MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Ohyama, Y.: Differential relations of theta functions. Osaka J. Math. 32(2), 431–450 (1995)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Ostrowski, A.: Über Dirichletsche Reihen und algebraische Differentialgleichungen. Math. Z. 8(3–4), 241–298 (1920).  https://doi.org/10.1007/BF01206530 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pogudin, G.: The primitive element theorem for differential fields with zero derivation on the base field. J. Pure Appl. Algebra 219(9), 4035–4041 (2015).  https://doi.org/10.1016/j.jpaa.2015.02.004 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Ritt, J.F.: Differential Equations from the Algebraic Standpoint. Colloquium Publications. American Mathematical Society, Providence (1932)CrossRefGoogle Scholar
  32. 32.
    Sánchez, O.L.: On the model companion of partial differential fields with an automorphism. Isr. J. Math. 212(1), 419–442 (2016).  https://doi.org/10.1007/s11856-016-1292-y MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Seidenberg, A.: Abstract differential algebra and the analytic case. II. In: Proceedings of the American Mathematical Society, vol. 23, no. 3, pp. 689–691, (1969). URL https://www.jstor.org/stable/2036611
  34. 34.
    Seidenberg, A.: Abstract differential algebra and the analytic case. In: Proceedings of the American Mathematical Society, vol. 9, no. 1, pp. 159–164, (1958). URL https://www.jstor.org/stable/2033416
  35. 35.
    Seidenberg, A.: Some basic theorems in differential algebra (characteristic p, arbitrary). Trans. Am. Math. Soc. 73(1), 174–190 (1952)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Singer, M.F.: The model theory of ordered differential fields. J. Symb. Logic 43(1), 82–91 (1978)MathSciNetCrossRefGoogle Scholar
  37. 37.
    van der Waerden, B.: Algebra. Springer, New York (1991)CrossRefGoogle Scholar
  38. 38.
    Wood, C.: Prime model extensions for differential fields of characteristic \(p \ne 0\). J. Symb. Logic 39(3), 469–477 (1974).  https://doi.org/10.2307/2272889 Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

Personalised recommendations