Hasse derivative
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
- [math]\displaystyle{ D^{(r)} X^n = \binom{n}{r} X^{n-r}, }[/math]
if n ≥ r and zero otherwise.[1] In characteristic zero we have
- [math]\displaystyle{ D^{(r)} = \frac{1}{r!} \left(\frac{\mathrm{d}}{\mathrm{d}X}\right)^r \ . }[/math]
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
- [math]\displaystyle{ D^{(r)}(fg) = \sum_{i=0}^r D^{(i)}(f) D^{(r-i)}(g) }[/math]
and an analogue of the chain rule.[2] Note that the [math]\displaystyle{ D^{(r)} }[/math] 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]
- [math]\displaystyle{ f = \sum_r D^{(r)}(f) \cdot t^r \ . }[/math]
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. https://books.google.com/books?id=d83YEA6ncsYC&q=%22Hasse+derivatives%22+OR+%22Hasse+derivative%22.
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