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 *X*^{n} is

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

## 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

and an analogue of the chain rule.^{[2]} Note that the 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]}

## References

- 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.