In mathematics, the Hasse derivative is 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 generalized derivation on k[X] and extends to a generalized derivation on the function field k(X), satisfying an analogue of the product rule
and an analogue of the chain rule. Note that the
are not themselves derivations in general, but are closely related.
A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety:
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