$$
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$$
Algebraic Faà di Bruno
The Faà di Bruno formula in the Taylor algebras E0040:
coverings in \(B_k\), partitions in \(A_k\), and the deformation
in \(C_k\) that interpolates between them.
(1) Theorem (Boolean Faà di Bruno in \(B_k\) and \(A_k\)). Let
\(q: \ik^e \to \ik^d\) and \(p: \ik^d \to \ik\) be polynomials,
\(x \in \ik^e\), \(v_1, \dots, v_k \in \ik^e\), \(y = q(x)\).
-
In \(B_k\) (coverings):
\[
\Delta(p \circ q; x; v_{[k]})
= \sum_{\substack{H \subseteq \KP_+([k]) \\
\bigcup H = [k]}}
\Delta(p; y; (\Delta(q; x; v_T))_{T \in H}).
\]
-
In \(A_k\) (partitions):
\[
D(p \circ q; x; v_{[k]})
= \sum_{\pi \in \Part([k])}
D(p; y; (D(q; x; v_B))_{B \in \pi}).
\]
Proof.
Write \(q_T := \Delta(q; x; v_T)\). By Taylor–Möbius duality
E0041 in \(B_k\),
\(q(x + \sum_i \delta_i v_i) = y + \sum_T q_T \delta^T\).
Since \((\delta^T)^2 = \delta^T\), the map
\(\delta_T \mapsto \delta^T\) defines an algebra map
\(B_l \to B_k\) with \(l = 2^k - 1\) directions \(q_T\). Applying
Taylor–Möbius duality to \(p\) at \(y\) via this map:
\((p \circ q)(x + \sum_i \delta_i v_i) =
p(y + \sum_T q_T \delta^T)\). Extracting \([\delta^{[k]}]\):
the covering condition \(\bigcup H = [k]\) comes from
\(\delta^{T_1} \cdots \delta^{T_r} = \delta^{T_1 \cup \cdots \cup T_r}\).
The same argument in \(A_k\): since \((\eps^T)^2 = 0\), the
product \(\eps^{T_1} \cdots \eps^{T_r} = 0\) unless the \(T_i\)
are pairwise disjoint. Coverings reduce to partitions.
Proof.
By Taylor–Möbius duality in \(C_k\) E0041,
\(c_S(t) = t^{-|S|} \Delta(p \circ q; x; (tv_i)_{i \in S})\).
The discrete covering formula, applied over \(\ik[t]\), expands
this as
\(t^{-|S|} \sum_{H \in \Cov(S)}
\Delta(p; y; (\Delta(q; x; (tv_i)_T))_{T \in H})\).
Each inner increment \(\Delta(q; x; (tv)_T)\) is divisible by
\(t^{|T|}\) by the \(C_k\) duality E0041. The outer difference
\(\Delta(p; y; w_1, \dots, w_r)\) vanishes on every hyperplane
\(w_j = 0\), so substituting \(w_j = t^{|T_j|} \tilde{q}_{T_j}\)
extracts \(t^{\mathrm{wt}(H)}\). At \(t = 0\), terms with
\(\mathrm{wt}(H) > |S|\) vanish; the remaining terms have
\(\mathrm{wt}(H) = |S|\), i.e. \(H \in \Part(S)\) E0008. For
a partition \(\pi\), the multilinear part of the substituted
difference is \(D(p; y; (D(q; x; v_B))_{B \in \pi})\) by
the multinomial theorem in \(A_k\) (nilpotent generators kill
all non-multilinear monomials).
Used by: