An Identity Theorem for the Fourier–Laplace Transform of Polytopes on Nonzero Complex Multiples of Rationally Parameterizable Hypersurfaces

Abstract

A set mathcal {S} of points in mathbb {R}^n is called a rationally parameterizable hypersurface if there is vector function varvec{sigma }:mathbb {R}^{n-1}rightarrow mathbb {R}^n having as components rational functions defined on some common domain D such that mathcal {S}={varvec{sigma }(textbf{t}):textbf{t}in D}. A generalized n-dimensional polytope in mathbb {R}^n is a union of a finite number of convex n-dimensional polytopes in mathbb {R}^n. The Fourier–Laplace transform of such a generalized polytope mathcal {P} in mathbb {R}^n is defined by F_{mathcal {P}}(textbf{z})=int _{mathcal {P}}e^{textbf{z}cdot textbf{x}},textbf{dx}. Let gamma be a fixed nonzero complex number. We prove that F_{mathcal {P}_1}(gamma varvec{sigma }(textbf{t}))=F_{mathcal {P}_2}(gamma varvec{sigma }(textbf{t})) for all textbf{t} in O implies mathcal {P}_1=mathcal {P}_2 if O is an open subset of D satisfying some well-defined conditions and we present similar results for the null set of the Fourier–Laplace transform of mathcal {P}. Moreover we show that this theorem can be applied to quadric hypersurfaces that do not contain a line, but at least two points, i.e., in particular to spheres.