Polynomial tests and a convergent local Fourier quotient

Exposition, proof, three examples, six solutions, renderer, exact geometry, three PNG/SVG drawings, learner orientation and reader presentation: GPT-6.1 Sol (OpenAI), Ultra, October 2026; CC0-1.0.

Human mathematical sources: Lars Hörmander, The Analysis of Linear Partial Differential Operators I, Definition7.3.5 and Lemma7.3.7, with the separate compact Fourier context Theorems7.3.1–7.3.2 and Lemma7.3.3. The complete LP10 source paragraph retains the exact approved programme locators. Bernard Malgrange’s approximation paper remains credited in receiving L011. No new external book-body reading, source-only closure, book prose or source media import is claimed. Human sources, Matplotlib, DejaVu Sans and MathJax retain their own rights and component terms.

The theorem supplies the local polynomial-annihilator statement and convergence of its quotient germ. Local-to-entire continuation, entire annihilators, compact singular-support hulls and ordinary support-hull arguments have their own stated hypotheses.