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