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Hasse derivative
In mathematics, the Hasse derivative is a derivation, 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
\( D^{(r)} X^n = \binom{n}{r} X^{n-r}, \ \)
if n ≥ r and zero otherwise.[1] In characteristic zero we have
\( D^{(r)} = \frac{1}{r!} \left(\frac{\mathrm{d}}{\mathrm{d}X}\right)^r \ . \)
Properties
The Hasse derivative is a derivation on k[X] and extends to a derivation on the function field k(X),[1] satisfying the product rule and the chain rule.[2]
A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety:[3]
\( f = \sum_r D^{(r)}(f) \cdot t^r \ . \)
References
Goldschmidt (2003) p.28
Goldschmidt (2003) p.29
Goldschmidt (2003) p.64
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.
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