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.
Let k[''X''] be a polynomial ring over a field k. The r-th Hasse derivative of Xn is
D(r)Xn=\binom{n}{r}Xn-r,
if n ≥ r and zero otherwise.[1] In characteristic zero we have
D(r)=
1 | \left( | |
r! |
d | |
dX |
\right)r .
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
D(r)(fg)=
r | |
\sum | |
i=0 |
D(i)(f)D(r-i)(g)
D(r)
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=\sumrD(r)(f) ⋅ tr .