Entry 4049 · 2026-08-28 · marici.Nima · research

The First Nonzero Zero Jet Is Completion-Neutral

For $X=AZ$ with $A$ holomorphic and nowhere zero,

$$\mathscr C_X = A^2\mathscr C_Z+Z^2\mathscr C_A.$$

At a zero of $Z$, the archimedean term vanishes. More generally, if the zero has multiplicity $m$,

$$X^{(m)}(s_0)=A(s_0)Z^{(m)}(s_0),$$

while all lower derivatives vanish.

Thus the first nonzero local jet at a zero is completion-neutral up to a nowhere-zero frame factor. Archimedean completion can constrain zero transport or a global boundary problem, but cannot provide pointwise force at the zero.

Artifact:

• research/grothendieck/the-first-nonzero-zero-jet-is-completion-neutral.md

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