claimroutine

Separated weights avoid center dependence

Chooses successive weights with

\prod_A w_a>C_*\prod_B v_b

, independent of the points

arithmetic weight separation used throughout the geometric proofs

Claim·§2.2 The curve inequality·inside thmInterpolation with separated weights

Statement (paper)

Fix a positive rational number

\sigma

satisfying , put

v_i=w_i/\theta

for

i>0

L_0=m+2

and

C_*=\max_{1\le k\le m}2k!(L_0/\sigma)^k

. Then the positive weights

w_1,\ldots,w_m

can be chosen successively, depending only on the fixed data

m,K,w_0,v_0,\theta

and the previously chosen weights (and not on any interpolation point), so that for every pair of subsets

A,B\subseteq\{0,\ldots,m\}

of the same cardinality between

1

and

m

\prod_{a\in A}w_a>C_*\prod_{b\in B}v_b

whenever the largest positive index in their symmetric difference belongs to

A

In plain English

Given a fixed rational

\sigma

satisfying , with

v_i=w_i/\theta

L_0=m+2

and

C_*=\max_{1\le k\le m}2k!(L_0/\sigma)^k

, the weights

w_1,\ldots,w_m

can be chosen successively (depending only on

m,K,w_0,v_0,\theta

and earlier weights, not on the points) so that

\prod_{a\in A}w_a>C_*\prod_{b\in B}v_b

for all equal-size subsets

A,B\subseteq\{0,\ldots,m\}

of size between

1

and

m

whenever the largest positive index in

A\triangle B

lies in

A

Why it's here

This is a purely arithmetic choice of weights, made by taking each new

w_i

large enough compared with the earlier ones. The resulting strict separation of weight products gives each normal-basis index set a dominant term, so the later curve-inequality arguments never depend on the interpolation points or on which center is involved.</br>

Classified routine:

This is a standard finite-dominance construction: choose each successive weight large enough that its appearance on the left dominates every relevant product of earlier weights. The choice depends only on fixed data and previously chosen weights, not on interpolation points.

Uses

Nothing else in the paper

Used by

claimProjective degree equals weighted curve degreeclaimIdeal order along a branch equals weighted contactclaimExcess contact yields auxiliary polynomialclaimPersistent component constrains normal basesclaimLogarithmic residues force constant YclaimFiber polynomial retains derivative vanishingclaimSeparated normals isolate one centerclaimSingle-center zero-pole count contradicts excess

Sources

PaperOpen

Section: §2.2 The curve inequality
Page: —
Label: —
Line: sections/interpolation.tex:180

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