nLab Hasse-Schmidt derivative




In a field of positive characteristic, the usual derivative of polynomials has bad properties. Let 𝕂\mathbb{K} be such a field of characteristic p>0p \gt 0. Consider the polynomial algebra 𝕂[X]\mathbb{K}[X]. The usual derivative of polynomials in one indeterminate defines a derivation 𝕂[X]𝕂[X]\mathbb{K}[X] \rightarrow \mathbb{K}[X] which associates PP' to PP.

We then have that (X p)=p.X p1=0(X^{p})'=p.X^{p-1}=0 but X p0X^{p} \neq 0 because 1 p=101^{p} = 1 \neq 0. We thus lack the property that P=0P' = 0 iff PP is a constant.

The notion of Hasse-Schmidt derivative comes as a replacement of the usual derivative and enjoys better properties.

It seems to be very linked to differential linear logic and to the notion of graded bimonoid.

Definition for one indeterminate

Suppose that RR is a commutative rig. For every polynomial PR[X]P \in R[X] and n0n \ge 0, we define the nthn-th Hasse-Schmidt derivative P (n)P^{(n)} of PP by:

  • P (0)=PP^{(0)} = P

and if n1n \ge 1:

  • (X n+p) (n)=[(n+pn).1 R]X p(X^{n+p})^{(n)} = \left[\binom{n+p}{n}.1_{R}\right] X^{p} if p0p \ge 0
  • (X k) (n)=0(X^{k})^{(n)} = 0 if n>kn \gt k
  • (aP+bQ) (n)=aP (n)+bQ (n)(aP+bQ)^{(n)} = aP^{(n)} + bQ^{(n)}

Definition for several indeterminates

Suppose that RR is a commutative rig. We look at the polynomials in R[X 1,...,X q]R[X_{1},...,X_{q}]. For every multiset Y 1...Y n{X 1,...,X q}Y_{1}...Y_{n} \in \{X_{1},...,X_{q}\} and PR[X 1,...,X q]P \in R[X_{1},...,X_{q}] we define the derivative D Y 1...Y n(P)D_{Y_{1}...Y_{n}}(P). Note that we treat in the same way all the higher-order derivatives and define them directly and not by induction. It is in this way directly clear that the order of the variables with respect to we derivate doesn’t matter.

Suppose that PP is a monic monomial, that is a multiset of variables in {X 1,...,X q}\{X_{1},...,X_{q}\}.

  • If Y 1...Y nY_{1}...Y_{n} doesn’t divide PP, then:

    D Y 1...Y n(P)=0 D_{Y_{1}...Y_{n}}(P) = 0
  • If Y 1...Y nY_{1}...Y_{n} divides PP, then:

    D Y 1...Y n(P)=[1iq X i|Y 1...Y n(nbroftimeX iinPnbroftimeX iinY 1...Y n).1 R]PY 1...Y n D_{Y_{1}...Y_{n}}(P) = \left[\underset{\substack{1 \le i \le q \\ X_{i} | Y_{1}...Y_{n}}}{\prod}\binom{nbr\;of\;time\;X_{i}\;in\;P}{nbr\;of\;time\;X_{i}\;in\;Y_{1}...Y_{n}}.1_{R}\right]\frac{P}{Y_{1}...Y_{n}}

We then prolongate by linearity:

  • D Y 1...Y n(aP+bQ)=aD Y 1...Y n(P)+bD Y 1...Y n(Q)D_{Y_{1}...Y_{n}}(aP+bQ) = aD_{Y_{1}...Y_{n}}(P) + bD_{Y_{1}...Y_{n}}(Q)

There is a very combinatorial interpretation which makes the link with the notion of graded bimonoid: if PP is a monic monomial, the derivative with respect to Y 1...Y nY_{1}...Y_{n} counts the number of ways to extract Y 1...Y nY_{1}...Y_{n} from PP and then multiplies it by PP where Y 1...Y nY_{1}...Y_{n} has been extracted. All the instances of a variable Y iY_{i} in PP must be considered as different entities in this counting.

For example there is (32)(41)\binom{3}{2}\binom{4}{1} ways to extract X 2YX^{2}Y from X 3Y 4X^{3}Y^{4}, we have (32)\binom{3}{2} choices of two instance of XX in X 3X^{3} and (41)\binom{4}{1} choices of two instances of YY in Y 4Y^{4}, thus:

D X 2Y(X 3Y 4)=(32)(41)XY 3=3.4.XY 3=12XY 3 D_{X^{2}Y}(X^{3}Y^{4}) = \binom{3}{2}\binom{4}{1}XY^{3} = 3.4.XY^{3} = 12XY^{3}


The notion was introduced in:

  • Helmut Hasse: Noch eine Begründung der Theorie der höheren Differrentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von F. K. Schmidt in Jena.), Reine Angew. Math. 177 (1937) 215-223 [paper]

The definition and basic properties are reviewed on this blog page:

  • Hasse derivative, Felix’ Math Place (2009), url

Last revised on August 18, 2022 at 18:35:50. See the history of this page for a list of all contributions to it.