= Hasse derivative
{c}
{title2=$f^{[\ell]}(x)=\sum_{j\geq\ell}\binom j\ell f_jx^{j-\ell}$}
{wiki}
The Hasse derivatives of a <polynomial> are the coefficients in $f(x+t)=\sum_{\ell\geq0}f^{[\ell]}(x)t^\ell$. A point $a$ has <multiplicity> at least $m$ exactly when $f^{[\ell]}(a)=0$ for $0\leq\ell<m$. Unlike iterated <formal derivatives> divided by <factorials>, this definition works in positive <characteristic>, where factorials can vanish. For example, in characteristic two $(x+1)^2$ has zero first <formal derivative> but nonzero second Hasse derivative.
// Target: coding-theory.bigb
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