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

$D^{(r)}{X}^n=(\genfrac{}{}{0ex}{}{n}{r})X^{n-r},$

if n ≥ r and zero otherwise. 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 generalized derivation on k[X] and extends to a generalized derivation on the function field k(X), satisfying an analogue of the product rule

$D^{(r)}(fg)=\sum _{i=0}^r{D}^{(i)}(f)D^{(r-i)}(g)$

and an analogue of the chain rule. Note that the $D^{(r)}$ 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:

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