In mathematics, the Hasse derivative is a generalisation of the

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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 truncation a ...

coordinate ring In algebraic geometry, an affine variety or affine algebraic variety is a certain kind of algebraic variety that can be described as a subset of an affine space. More formally, an affine algebraic set is the set of the common zeros over an algeb ...

s of

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...

Definition

Let ''k'' 'X''be a

polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another 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 \ .

Notes

References

* Differential algebra {{algebra-stub