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.
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Definition
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
Properties
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:
References
Hasse derivative Wikipedia(Text) CC BY-SA