On a theory of the b-function in positive characteristic

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Abstract

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 

Notes

Acknowledgements

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 research presented here was made while the author was a C.L.E. Moore instructor at MIT and a research fellow at the Higher School of Economics, Moscow. The author was partially supported by EPSRC grant EP/L005190/1.

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Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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