Hasse Derivative
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Hasse Derivative

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

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

Properties

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]

References

  1. ^ a b Goldschmidt (2003) p.28
  2. ^ Goldschmidt (2003) p.29
  3. ^ Goldschmidt (2003) p.64
  • Goldschmidt, David M. (2003). Algebraic functions and projective curves. Graduate Texts in Mathematics. 215. New York, NY: Springer-Verlag. ISBN 0-387-95432-5. Zbl 1034.14011.

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