 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-129/4/b/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

For

h (x) = e_{p} (q (x))

, the product in the

U^{3}

e_{p} (\sum_{ϵ \in {0, 1}^{3}} (- 1)^{∣ ϵ ∣} q (x + ϵ \cdot h)) .

The expression in parentheses is the third additive derivative of the quadratic form

q

, so it vanishes. Every cube contributes one and therefore the quadratic phase satisfies

∥ h ∥_{U^{3}} = 1.

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