Hasse derivative

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.

Definition

Let k[X] be a polynomial ring over a field k. The r-th Hasse derivative of Xⁿ is

D^{{(r)}}X^{n}={\binom {n}{r}}X^{{n-r}},\

if n ≥ r and zero otherwise.^[1] In characteristic zero we have

D^{{(r)}}={\frac {1}{r!}}\left({\frac {{\mathrm {d}}}{{\mathrm {d}}X}}\right)^{r}\ .

Properties

The Hasse derivative is a derivation on k[X] and extends to a derivation on the function field k(X),^[1] satisfying the product rule and the chain rule.^[2]

A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety:^[3]

f=\sum _{r}D^{{(r)}}(f)\cdot t^{r}\ .

References

1 2 Goldschmidt (2003) p.28
↑ Goldschmidt (2003) p.29
↑ 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.

This article is issued from Wikipedia - version of the 6/20/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.