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 Xn is
D ( r ) X n = ( n r ) X n − r , {\displaystyle D^{(r)}X^{n}={\binom {n}{r}}X^{n-r},}
if n ≥ r and zero otherwise. In characteristic zero we have
D ( r ) = 1 r ! ( d d X ) r . {\displaystyle D^{(r)}={\frac {1}{r!}}\left({\frac {\mathrm {d} }{\mathrm {d} X}}\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 ( r ) ( f g ) = ∑ i = 0 r D ( i ) ( f ) D ( r − i ) ( g ) {\displaystyle D^{(r)}(fg)=\sum _{i=0}^{r}D^{(i)}(f)D^{(r-i)}(g)}
and an analogue of the chain rule. Note that the D ( r ) {\displaystyle D^{(r)}} 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:
f = ∑ r D ( r ) ( f ) ⋅ t r . {\displaystyle f=\sum _{r}D^{(r)}(f)\cdot t^{r}\ .}
Notes
References Goldschmidt, David M. (2003). Algebraic functions and projective curves. Graduate Texts in Mathematics. Vol. 215. New York, NY: Springer-Verlag. ISBN 0-387-95432-5. Zbl 1034.14011.