On a theory of the b-function in positive characteristic

Journal article

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 $Z_p$. The zero-locus of the $b$-function is thus naturally interpreted as a subset of $Z_p$, which we call the set of roots of $b_f$. We prove that bf 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 $f$.

Authors: Bitoun, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer Verlag
• Journal: Selecta Mathematica
• Publication date: 2018-02-02
• Acceptance date: 2017-12-12
• DOI: 10.1007/s00029-017-0383-x
• EISSN: 1420-9020
• ISSN: 1022-1824