  Carath\'eodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be
written in the form $c_p=\sum_{j=1}^m \rho_j {\epsilon_j}^p$ with $p=1,...,n$,
where the $\epsilon_j$s are different unimodular complex numbers, the $\rho_j$s
are strictly positive numbers and integer $m$ never exceeds $n$. We give the
conditions to be obeyed for the former property to hold true if the $\rho_j$s
are simply required to be real and different from zero. It turns out that the
number of the possible choices of the signs of the $\rho_j$s are {at most}
equal to the number of the different eigenvalues of the Hermitian Toeplitz
matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the
complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for
neutron scattering. Its proof is made possible by a lemma - which is an
interesting side result - that establishes a necessary and sufficient condition
for the unimodularity of the roots of a polynomial based only on the polynomial
coefficients.
  Keywords: Toeplitz matrix factorization, unimodular roots, neutron
scattering, signal theory, inverse problems.
  PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga
