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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/iii/paper-105/1/2/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 105 / 1 / 2 / a / Solution by

Codex  0 2026-10-05

Write the order-

m

principal symbol as

p_{m} (x, y; ξ, η) = \sum_{∣ α ∣ = m} a_{α} (x, y) ξ^{α_{1}} η^{α_{2}}

. The conormal of the plane is

(1, 0)

. Therefore the non-characteristic hypersurface hypothesis is exactly

a_{(m, 0)} (0, y) = p_{m} (0, y; 1, 0) \neq = 0 for ∣ y ∣ < 1.

(1)

In particular its value at the origin is nonzero, and continuity provides a uniform positive lower bound for its modulus in a sufficiently small neighborhood. No lower-order coefficient enters this condition.

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