Manuscript X-B-ST@1 · X-B-ST.tex · lemma · d03c150c

Lemma: Degree-one part of the ideal

C3: normalization and degree-one uniqueness › The equation ideal and separation by solutions · lines 1949–1977 · proof included or linked

X-B-ST@1#lem:graded, X-B-ST@1#eq:linearzero

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The pointwise ideal $J_{0}$ is graded by field degree, and

J_{0} \cap A_{off}^{(1)} = {F_{P_{m} a} : a \in D} .

Hence

ϕ (d) = 0 ⟺ d \in P_{m} D .

Proof. An ideal element is a finite sum $\sum_{i} G_{i} F_{P_{m} a_{i}}$ . Decompose each polynomial $G_{i}$ into its homogeneous parts. Each homogeneous component of the sum is still in the ideal. Its degree-one part consists exactly of constant multiples of the linear generators, hence is $F_{P_{m} a}$ for some compact $a$ .

The equality $F_{d} = F_{P_{m} a}$ implies $d = P_{m} a$ , since the functionals can be tested on arbitrary compact smooth configurations.

Displayed formulas in this passage

unlabeled · line 1953
$J_{0} \cap A_{off}^{(1)} = {F_{P_{m} a} : a \in D} .$
X-B-ST@1#eq:linearzero · line 1958
$ϕ (d) = 0 ⟺ d \in P_{m} D .$

Lemma: Degree-one part of the ideal

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