Locally polynomially integrable surfaces and finite stationary phase expansions

Let M be a strictly convex smooth connected hypersurface in ℝⁿ and M^ its convex hull. We say that M is locally polynomially integrable if for every point a ∈ M the (n − 1)-dimensional volume of the cross-section of M^ by a parallel translation of the tangent hyperplane at a to a small distance t depends polynomially on t. It is conjectured that only quadrics in odd-dimensional spaces possess such a property. The main result of this article partially confirms the conjecture. The study of integrable domains and surfaces is motivated by a conjecture of V. I. Arnold about algebraically integrable domains. The result and the proof are related to studying oscillating integrals for which the asymptotic stationary phase expansions consist of a finite number of terms.