E0043 proposition notation E0001

Iterated algebraic Faà di Bruno

The algebraic Faà di Bruno E0042 extends to \(m\)-fold compositions of polynomial maps, indexed by higher partitions \(\Part_m(k)\) E0007 in \(A_k\).

(1) Proposition (Iterated infinitesimal Faà di Bruno). Let \(f_1, \dots, f_m\) be polynomial maps with \(f_r: \ik^{d_{r-1}} \to \ik^{d_r}\), \(x \in \ik^{d_0}\), \(v_1, \dots, v_k \in \ik^{d_0}\), and \(x_r = (f_r \circ \cdots \circ f_1)(x)\).

  • Boolean form:

    \[ D(f_m \circ \cdots \circ f_1; x; v_\bullet) = \sum_{H \in \Part_m(k)} D^H(f_1, \dots, f_m; x; v_\bullet). \]

  • Binomial form. For \(\gamma \in \IN_0^k\):

    \[ D^\gamma(f_m \circ \cdots \circ f_1; x; v_\bullet) = \sum_{\kappa \in \KM_+^m(k)} \Part_m(\gamma, \kappa)\, D^\kappa(f_1, \dots, f_m; x; v_\bullet). \]

Here \(D^H\) and \(D^\kappa\) are the iterated differentials E0021 for polynomial maps, and \(\Part_m(\gamma, \kappa) = \#\set{H \in \Part_m(S(\gamma)) : \nu(H) = \kappa}\).

Proof. Ad Boolean) Evaluate \(f_m \circ \cdots \circ f_1\) on the infinitesimal cube \((x + \sum_i \eps_i v_i)\) in \(A_k\) by composing \(m\) Taylor–Möbius expansions E0041. At each level, the nilpotent multiplication \(\eps^{T_1} \cdots \eps^{T_p} = 0\) unless the \(T_i\) are pairwise disjoint forces partition logic: only higher partitions \(H \in \Part_m(k)\) contribute. Reading off the \(\eps^{[k]}\)-component gives the scalar identity.

Ad Binomial) Apply the Boolean form to \(S(\gamma)\) with \(|\gamma|\) directions where \(v_i\) is repeated \(\gamma_i\) times. Since \(D^H\) depends only on the profile \(\kappa = \nu(H)\) by symmetry of the derivatives, grouping by profile gives the stated sum with coefficient \(\Part_m(\gamma, \kappa)\).

(2) Proposition (Partition coefficient recursion). \(\Part_m(\gamma, \kappa) = 0\) unless \(\kappa \vdash \gamma\), and \(\Part_1(\gamma, \alpha) = \delta_{\gamma, \alpha}\). For \(m \geq 2\) and \(\kappa \vdash \gamma\):

\[ \Part_m(\gamma, \kappa) = \frac{\gamma!}{\kappa!} \prod_{\beta \in \supp(\kappa)} \left(\frac{\Part_{m-1}(\lf(\beta), \beta)} {\lf(\beta)!}\right)^{\kappa(\beta)}. \]

The first closed case is \(\Part_2(\gamma, \kappa) = \gamma! / (\kappa! \prod_\alpha (\alpha!)^{\kappa(\alpha)})\).

Proof. An \(m\)-fold partition with profile \(\kappa\) is built in two steps: (1) partition \(S(\gamma)\) into blocks matched with \(\supp(\kappa)\), where a block matched with \(\beta\) has type \(\lf(\beta)\) — there are \(\gamma! / (\kappa! \prod_\beta (\lf(\beta)!)^{\kappa(\beta)})\) such matched partitions; (2) equip each block with an \((m-1)\)-fold partition of profile \(\beta\), in \(\Part_{m-1}(\lf(\beta), \beta)\) ways. Multiplying gives the recursion.