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

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

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

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

Properties

The Hasse derivative is a generalized derivation on k[X] and extends to a generalized derivation on the function field k(X),^[1] satisfying an analogue of the product rule

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

and an analogue of the chain rule.^[2] Note that the ${\displaystyle D^{(r)}}$ 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:^[3]

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