In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of

Taylor's theorem In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...

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 :

D^ X^n = \binom X^,

if ''n'' ≥ ''r'' and zero otherwise.Goldschmidt (2003) p.28 In characteristic zero we have :

D^ = \frac \left(\frac\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^(fg) = \sum_^r D^(f) D^(g)

and an analogue of the chain rule.Goldschmidt (2003) p.29 Note that the

D^

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:Goldschmidt (2003) p.64 :

f = \sum_r D^(f) \cdot t^r \ .

References

* Differential algebra {{algebra-stub