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. [1] In characteristic zero we have
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. [1] 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), [1] satisfying an analogue of the product rule
and an analogue of the chain rule. [2] 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: [3]
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