  We show that a monic univariate polynomial over a field of characteristic
zero, with $k$ distinct non-zero known roots, is determined by its $k$ proper
leading coefficients by providing an explicit algorithm for computing the
multiplicities of each root. We provide a version of the result and
accompanying algorithm when the field is not algebraically closed by
considering the minimal polynomials of the roots. Furthermore, we show how to
perform the aforementioned algorithm in a numerically stable manner over
$\mathbb{C}$, and then apply it to obtain new characteristic polynomials of
hypergraphs.
