Hasse derivative

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In mathematics, the Hasse derivative is a derivation, a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties.


Let k[X] be a polynomial ring over a field k. The r-th Hasse derivative of Xn is

if nr and zero otherwise.[1] In characteristic zero we have


The Hasse derivative is a derivation on k[X] and extends to a derivation on the function field k(X),[1] satisfying the product rule and the chain rule.[2]

A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety:[3]


  1. ^ a b Goldschmidt (2003) p.28
  2. ^ Goldschmidt (2003) p.29
  3. ^ Goldschmidt (2003) p.64