The Hasse derivatives of a polynomial are the coefficients in . A point has multiplicity at least exactly when for . Unlike iterated formal derivatives divided by factorials, this definition works in positive characteristic, where factorials can vanish. For example, in characteristic two has zero first formal derivative but nonzero second Hasse derivative.
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The Hasse derivative is a mathematical concept used primarily in the context of p-adic analysis and algebraic geometry, particularly within the study of p-adic fields and formal power series. It is named after the mathematician Helmut Hasse. In simple terms, the Hasse derivative can be thought of as a form of differentiation that is adapted to p-adic contexts, similar to how we differentiate functions in classical calculus.