  We prove that any finite union $P$ of interior-disjoint polytopes in
${\mathbb R}^d$ has the Pompeiu property, a result first proved by Williams
[Wil76]. This means that if a continuous function $f$ on $R^d$ integrates to 0
on any congruent copy of $P$ then $f$ is identically 0. By a fundamental result
of Brown, Schreiber and Taylor [BST73] this is equivalent to showing that the
Fourier-Laplace transform of the indicator function of $P$ does not vanish
identically on any 0-centered complex sphere in ${\mathbb C}^d$ . Our proof
initially follows the recent one of Machado and Robins [MR23] who are using the
Brion-Barvinok formula for the Fourier-Laplace transform of a polytope. But we
simplify this method considerably by removing the use of properties of Bessel
function zeros. Instead we use some elementary arguments on the growth of
linear combinations of exponentials with rational functions as coefficients.
Our approach allows us to prove the non-existence of complex spheres of any
center in the zero-set of the Fourier-Laplace transform. The planar case is
even simpler in that we do not even need the Brion-Barvinok formula. We then go
further in the question of which sets can be contained in the null set of the
Fourier-Laplace transform of a polytope by extending results of Engel [Eng23]
who showed that rationally parametrized hypersurfaces, under some mild
conditions, cannot be contained in this null-set. We show that a rationally
parametrized curve which is not contained in an affine hyperplane in ${\mathbb
C}^d$ cannot be contained in this null-set. Results about curves parametrized
by meromorphic functions are also given.
