lemmaroutineunresolved_ref

Local multiplicity comparison of weights and costs

Bounds ∏ρa≤2k! ε−k∏κb\prod\rho_a\le 2k!\,\varepsilon^{-k}\prod\kappa_b for normal bases of a component
constrains normal bases in the curve inequality proof
Lemma·geom:multiplicity·§2.1 A local multiplicity comparison

Statement (paper)

Give the coordinates x1,…,xdx_1,\ldots,x_d positive rational degree weights ρ1,…,ρd\rho_1,\ldots,\rho_d. On an open subset of Cd\CC^d, let D1,…,DdD_1,\ldots,D_d be commuting analytic vector fields forming a frame, and assign them positive rational costs κ1,…,κd\kappa_1,\ldots,\kappa_d. Fix ε>0\varepsilon>0. Let N>0N>0. Suppose that polynomials f1,…,frf_1,\ldots,f_r have weighted degree at most NN, that ZZ is an irreducible component of their common zero set of codimension k≥1k\ge1, and that Dγfℓ∣Z=0whenever∑bκbγb<εN. D^\gamma f_\ell\big|_Z=0 \quad\text{whenever}\quad \sum_b\kappa_b\gamma_b<\varepsilon N. For any coordinate normal basis AA and any DD-field normal basis BB, each of cardinality kk, one has ∏a∈Aρa≤2k! ε−k∏b∈Bκb \prod_{a\in A}\rho_a \le 2k!\,\varepsilon^{-k}\prod_{b\in B}\kappa_b The bound is uniform in the equations, the component ZZ, and the analytic frame; no lower threshold for NN is required.

In plain English

If polynomials of weighted degree at most NN have all DD-derivatives of cost below εN\varepsilon N vanishing on a codimension-kk component ZZ of their common zeros, then for any coordinate normal basis AA and DD-field normal basis BB of size kk, ∏a∈Aρa≤2k! ε−k∏b∈Bκb\prod_{a\in A}\rho_a\le 2k!\,\varepsilon^{-k}\prod_{b\in B}\kappa_b. The bound is uniform in the equations, ZZ and the frame, and holds for every N>0N>0.

Why it's here

The proof compares two bounds on the local intersection length at a smooth point of ZZ on a coordinate slice. The vanishing derivatives give a lower bound of order (εN)k/∏κb(\varepsilon N)^k/\prod\kappa_b, and weighted Bezout gives an upper bound Nk/∏ρaN^k/\prod\rho_a. Cancelling NkN^k yields the inequality, which the later curve-inequality arguments use to constrain normal bases of persistent components.
Classified routine:
This combines standard local intersection-multiplicity bounds: derivative vanishing gives a lower bound on length, while weighted Bézout gives an upper bound. Comparing the two yields the stated inequality.

Uses

Nothing else in the paper

Used by

propCurve inequality for weighted degreeclaimPersistent component constrains normal basesclaimSeparated normals isolate one center

Inside the proof

claimPersistent derivatives force large local lengthclaimWeighted Bezout caps slice intersection length

Sources

PaperOpen
Section
§2.1 A local multiplicity comparison
Page
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Label
geom:multiplicity
Line
sections/interpolation.tex:72
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Phase 2 aligns this node to a Lean declaration.

Friction

  • warn
    unresolved_ref
    The slice-length upper bound uses polynomial Bézout with isolated local multiplicities, cited externally at interpolation.tex:162–167.