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. 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.