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 n ≥ r and zero otherwise. 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), satisfying the product rule and the chain rule.
A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety:
- ^ a b Goldschmidt (2003) p.28
- ^ Goldschmidt (2003) p.29
- ^ Goldschmidt (2003) p.64