E0034 theorem notation E0020 → E0001

Higher symbols and curved collapse

The collapse theorem E0031 extracts the leading term of \(t^{-k} \Delta(f; x; \lambda_t c)\). The higher symbols \(\sigma_{k,m}\) capture the full asymptotic expansion to all orders, with \(\sigma_{k,0} = \sigma_k\) E0032.

(1) Definition (Higher symbols). For a cover \(\KC \in \Cov(k)\) and multiplicities \(\nu: \KC \to \IN_{>0}\), set \(\mathrm{wt}(\nu) := \sum_{A \in \KC} |A| \nu_A\) and \(\nu! := \prod_{A \in \KC} \nu_A!\). For \(m \geq 0\):

\[ \sigma_{k,m}(c) := \sum_{\KC \in \Cov(k)} \sum_{\substack{\nu: \KC \to \IN_{>0} \\ \mathrm{wt}(\nu) = k + m}} \frac{1}{\nu!} \prod_{A \in \KC} c(A)^{\nu_A} \in ST_{k+m}(X; x). \]

The multiplicity \(\nu_A\) records how many times the direction \(c(A)\) occurs. Minimal weight (\(m = 0\)) forces every \(\nu_A = 1\) and every block disjoint — recovering the first-order symbol \(\sigma_k\) E0032.

(2) Theorem (Curved collapse). Let \(M \geq 0\) and let \(f\) be \(C^{k+M}\) near \(x\). Then

\[ \Delta(f; x; \lambda_t c) = \sum_{m=0}^{M} t^{k+m}\, D(f; x; \sigma_{k,m}(c)) + t^{k+M}\, R_{k,M}(f; x; c, t), \]

where \(\sup_{c \in B} \|R_{k,M}(f; x; c, t)\| \to 0\) as \(t \to 0\) for every bounded \(B \subseteq CT_k(X; x)\).

Proof. By affine reconstruction E0018, \(\Delta(f; x; \lambda_t c) = \sum_{\KC \in \Cov(k)} \Delta(f; x; (t^{|A|} c(A))_{A \in \KC})\). For a fixed cover \(\KC\), expand the affine difference as an alternating sum over \(J \subseteq \KC\) and apply the Taylor formula with Peano remainder to \(f(x + \sum_{A \in J} t^{|A|} c(A))\).

In the Taylor polynomial, the multinomial indexed by \(\nu: \KC \to \IN_0\) has coefficient \(1/\nu!\) and contributes \(t^{\mathrm{wt}(\nu)} D(f; x; \prod_A c(A)^{\nu_A})\). Its coefficient in the alternating sum is \(\sum_{\supp \nu \subseteq J \subseteq \KC} (-1)^{|\KC| - |J|} = [\supp \nu = \KC]\) by the Boolean sieve E0002. Only multiplicities with every \(\nu_A > 0\) survive.

Group the surviving terms by weight: weights \(k, k+1, \dots, k+M\) give \(\sigma_{k,0}, \dots, \sigma_{k,M}\). Terms of weight \(> k + M\) in the Taylor polynomial are \(O(t^{k+M+1})\), and the Peano remainders are \(o(t^{k+M})\), both uniformly for bounded \(c\). Their sum divided by \(t^{k+M}\) defines \(R_{k,M}\).

(3) Corollary (Affine collapse). On affine cubes, the higher symbols specialize to multi-index sums:

\[ \sigma_{k,m}(\Aff(v_1, \dots, v_k)) = \sum_{\substack{\nu \in \IN_{>0}^k \\ \mathrm{wt}(\nu) = k + m}} \frac{1}{\nu!}\, v^\nu, \]

where \(v^\nu = v_1^{\nu_1} \cdots v_k^{\nu_k}\) in the symmetric algebra. For \(m = 0\), only \(\nu = (1, \dots, 1)\) contributes, giving \(v_1 \cdots v_k\).

Proof. For an affine cube, \(c(\set{i}) = v_i\) and \(c(A) = 0\) for \(|A| \geq 2\). Only covers by singletons contribute (\(\KC = \set{\set{1}, \dots, \set{k}}\)), and the multiplicity \(\nu\) assigns \(\nu_i = \nu(\set{i}) \geq 1\) to each singleton. The weight is \(\mathrm{wt}(\nu) = \sum_i \nu_i = k + m\) and \(c(\set{i})^{\nu_i} = v_i^{\nu_i}\).