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.

Definition

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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

 

Properties

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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]

 

Notes

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

References

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  • Goldschmidt, David M. (2003). Algebraic functions and projective curves. Graduate Texts in Mathematics. Vol. 215. New York, NY: Springer-Verlag. ISBN 0-387-95432-5. Zbl 1034.14011.