On a theory of the b-function in positive characteristic

• DOI: 10.1007/s00029-017-0383-x
• Publication Date: 2018-02-01T21:20:59Z
• Authors: Thomas Bitoun
• Keywords: Mathematics - Algebraic Geometry; FOS: Mathematics; 13A35, 14F10; 0101 mathematics; Algebraic Geometry (math.AG); 01 natural sciences

Final version<br/>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 Musta���� 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 $f.$<br/>