On a theory of the b-function in positive characteristic

Open Access


We present a theory of the b-function (or Bernstein–Sato polynomial) in positive characteristic. Let f be a non-constant polynomial with coefficients in a perfect field k of characteristic \(p>0.\) Its b-function \(b_f\) is defined to be an ideal of the algebra of continuous k-valued functions on \({\mathbb {Z}}_p.\) The zero-locus of the b-function is thus naturally interpreted as a subset of \({\mathbb {Z}}_p,\) which we call the set of roots of \(b_f.\) We prove that \(b_f\) has finitely many roots and that they are negative rational numbers. Our construction builds on an earlier work of Mustaţă and is in terms of D-modules, where D is the ring of Grothendieck differential operators. We use the Frobenius to obtain finiteness properties of \(b_f\) and relate it to the test ideals of f.

Mathematics Subject Classification

13A35 14F10 



This theory of the b-function in characteristic p sprang from my attempt to understand [20]. I am indebted to Mircea Mustaţă for the work done there. I would also like to thank him for initially mentioning the problem and answering many questions about test ideals. I thank Konstantin Ardakov and Francesco Baldassarri for introducing me to Mahler’s Theorem, thus clarifying the appearance of \({\mathbb {Z}}_p\) in the theory. Shunsuke Takagi kindly pointed me towards Example 3.0.8. Finally I would like to thank Roman Bezrukavnikov and Pavel Etingof for interesting discussions at the very beginning of this project. The author was partially supported by EPSRC Grant EP/L005190/1.


  1. 1.
    Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Co., Reading (1969)MATHGoogle Scholar
  2. 2.
    Beĭlinson, A.A.: How to glue perverse sheaves. In: \(K\)-Theory, Arithmetic and Geometry (Moscow, 1984–1986), vol. 1289 of Lecture Notes in Math., pp. 42–51. Springer, Berlin (1987)Google Scholar
  3. 3.
    Bernšteĭn, I.N.: Analytic continuation of generalized functions with respect to a parameter. Funkcional. Anal. i Priložen. 6(4), 26–40 (1972)MathSciNetGoogle Scholar
  4. 4.
    Bernstein, J.: Algebraic theory of D-modules. Available at
  5. 5.
    Berthelot, P.: \(D\)-modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. (4) 29(2), 185–272 (1996)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Blickle, M., Mustaţǎ, M., Smith, K.E.: Discreteness and rationality of \(F\)-thresholds. Michigan Math. J. 57, 43–61 (2008). Special volume in honor of Melvin HochsterMathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Blickle, M., Mustaţă, M., Smith, K.E.: \(F\)-thresholds of hypersurfaces. Trans. Am. Math. Soc. 361(12), 6549–6565 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ein, L., Lazarsfeld, R., Smith, K.E., Varolin, D.: Jumping coefficients of multiplier ideals. Duke Math. J. 123(3), 469–506 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Emerton, M., Kisin, M.: The Riemann–Hilbert correspondence for unit \(F\)-crystals. Astérisque 293, vi+257 (2004)MathSciNetMATHGoogle Scholar
  10. 10.
    Granville, A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. In: Organic Mathematics (Burnaby, BC, 1995), vol. 20 of CMS Conf. Proc., pp. 253–276. Amer. Math. Soc., Providence, RI (1997)Google Scholar
  11. 11.
    Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV. Inst. Hautes Études Sci. Publ. Math. 32, 361 (1967)MATHGoogle Scholar
  12. 12.
    Hara, N., Yoshida, K.-I.: A generalization of tight closure and multiplier ideals. Trans. Am. Math. Soc. 355(8), 3143–3174 (2003). (electronic)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kashiwara, M.: \(B\)-functions and holonomic systems. Rationality of roots of \(B\)-functions. Invent. Math. 38(1):33–53 (1976/77)Google Scholar
  14. 14.
    Kashiwara, M.: \(D\)-modules and microlocal calculus, vol. 217 of translations of mathematical monographs. American Mathematical Society, Providence, RI, 2003. Translated from the 2000 Japanese original by Mutsumi Saito, Iwanami Series in Modern MathematicsGoogle Scholar
  15. 15.
    Kollár, J.: Singularities of pairs. In: Algebraic Geometry—Santa Cruz 1995, vol. 62 of Proc. Sympos. Pure Math., pp. 221–287. Amer. Math. Soc., Providence, RI (1997)Google Scholar
  16. 16.
    Lazard, M.: Groupes analytiques \(p\)-adiques. Inst. Hautes Études Sci. Publ. Math. 26, 389–603 (1965)MathSciNetMATHGoogle Scholar
  17. 17.
    Leykin, A., Walther, U.: Survey on the \( d \)-module \( f^s\). arXiv preprint arXiv:1504.07516 (2015)
  18. 18.
    Lyubeznik, G.: \(F\)-modules: applications to local cohomology and \(D\)-modules in characteristic \(p>0\). J. Reine Angew. Math. 491, 65–130 (1997)MathSciNetMATHGoogle Scholar
  19. 19.
    Malgrange, B.: Polynômes de Bernstein–Sato et cohomologie évanescente. In: Analysis and Topology on Singular Spaces, II, III (Luminy, 1981), vol. 101 of Astérisque, pp. 243–267. Soc. Math. France, Paris (1983)Google Scholar
  20. 20.
    Mustaţă, M.: Bernstein–Sato polynomials in positive characteristic. J. Algebra 321(1), 128–151 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mustaţǎ, M., Takagi, S., Watanabe. K.: F-thresholds and Bernstein–Sato polynomials. In: European Congress of Mathematics, pp 341–364. Eur. Math. Soc., Zürich (2005)Google Scholar
  22. 22.
    Saito, M.: Introduction to a theory of \(b\)-functions. arXiv preprint arXiv:math/0610783 (2006)
  23. 23.
    Smith, S.P.: The global homological dimension of the ring of differential operators on a nonsingular variety over a field of positive characteristic. J. Algebra 107(1), 98–105 (1987)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations